**3. Communication theoretical limits, coding and signal processing**

Both multiconductor interconnects and wireless multiantenna interconnects can be interpreted as discrete-time, multi-input-multi-output (MIMO) systems. Such systems have been subject to extensive study in the recent past in the field of digital, especially mobile communications. Starting from the analysis of their promising information theoretic capabilities (e.g., [41]), a large amount of signal processing and coding techniques have been developed, that aim at achieving the information theoretic bounds (e.g., [42–44, 46]).

The common approach to handle spatio-temporal interference in MIMO systems, involves either linear or non-linear transmit and receive signal processing, which job is to transform the original MIMO system into a »virtual« MIMO system, where large amounts of spatio-temporal interference have been removed [48]. All state of the art MIMO signal processing techniques have in common that they assume that either the receiver, the transmitter, or both, have access to, or can generate signals with arbitrary precision. This implies, in practice, the existence of ADC and DAC components with a large enough resolution such that the non-linear effects of signal quantization can be neglected. However, in multiconductor, or wireless

10 Will-be-set-by-IN-TECH 84 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Chip-to-Chip and On-Chip Communications <sup>11</sup>

Here, x<sup>∞</sup>

defined as

∑*L*

<sup>0</sup> and <sup>r</sup>*t*−<sup>1</sup>

cardinality |X | <sup>=</sup> |R| <sup>=</sup> <sup>4</sup>*N*.

complexity and power dissipation.

*C* uniform

*Ms* = max

*Pij*:(*i*,*j*)∈T

than this *uniform capacity* (*Pi*,*<sup>j</sup>* takes 0 or 2−*K*), i.e., *C* uniform

optimization to homogeneous Markov sources.

This coding approach incorporates the following four ideas:

*3.1.1. Code-design*

<sup>1</sup> stand for the sequences [x−*L*, *<sup>x</sup>*−*L*+1,...] and [r1, <sup>r</sup><sup>2</sup> ..., <sup>r</sup>*t*−1], respectively.

[y*t*]*c*,*<sup>i</sup>* · [y¯]*c*,*<sup>i</sup> ση*/ √2

signal energy is normalized to 1, that is �x*t*�<sup>2</sup> = 1. The signal-to-noise ratio is accordingly

SNR = 1/*σ*<sup>2</sup>

*N* ∏ *i*=1 Φ 

Here, [x*t*]**R**,*<sup>i</sup>* ([x*t*]**I**,*i*) denotes the *i*-th real (imaginary) component of the input vector x*t*, ¯y =

cumulative normal distribution function. The input symbols are modulated using QPSK (or BPSK with 1-bit DAC). Consequently, we consider ISI channels with the input and output

We are looking for codes which maximally increase throughput and that allow fast decoding with good performance. In this way, the code design has to focus on finding low-complexity codes providing good coding gain while having low overhead both with respect to circuit

Even though linear block and convolution coding schemes are favorable candidates for error correction, they are not able to decrease power consumption and eliminate the residual error floor caused by the crosstalk even in the noiseless case [31]. This is due to their structural

Therefore, the coding schemes which are needed are non-linear and – for having good performance and, at the same time, a low complexity – for instance have memory of order one. In [51–53], an information-theoretic framework was developed as a practical design guideline

*Sk* = <sup>x</sup>*<sup>k</sup>*

<sup>I</sup>(x; <sup>y</sup>) s.t. *Pij* <sup>=</sup> Pr(*Sk* <sup>=</sup> *<sup>j</sup>*|*Sk*−<sup>1</sup> <sup>=</sup> *<sup>i</sup>*) ∈ {0, 2−*K*},

*<sup>k</sup>*−*Ms*+<sup>1</sup> ∈ X *Ms*,

*Ms* ≤ *CMs*.

*Ms* ≥ *L*, (6)

for novel codes. To this end, the following optimization has been considered in [52]

where *K* defines the rate *R* = *K*/(2*N*) of the code (for QPSK modulation) and *C* uniform

maximum channel capacity that can be attained with a homogeneous Markov source of order *Ms*. In general, the capacity of an unconstrained Markov source of order *Ms* [30] is higher

1. In order to avoid the complexity of maximizing an arbitrary Markov source, we restrict the

*<sup>t</sup>* ] = *<sup>σ</sup>*<sup>2</sup>

Chip-to-Chip and On-Chip Communications 85

*<sup>η</sup>* . (4)

1 2*π x* <sup>−</sup><sup>∞</sup> *<sup>e</sup>*<sup>−</sup> *<sup>t</sup>*<sup>2</sup>

*<sup>η</sup>* I*N*. The transmit

. (5)

<sup>2</sup> *dt* is the

*Ms* is the

The noise is additive white Gaussian with covariance matrix E[n*t*n<sup>H</sup>

*<sup>t</sup>*−*L*) = ∏

*<sup>k</sup>*=<sup>0</sup> <sup>H</sup>*k*x*t*−*<sup>k</sup>* is the noise-free unquantized receive vector and <sup>Φ</sup>(*x*) = <sup>√</sup>

properties (linear codes) and the coarse quantization of the channel.

*c*∈{**R**,**I**}

The channel transition probabilities can be calculated via

Pr(r*t*|x*<sup>t</sup>*

**Figure 10.** MIMO channel with ISI and single-bit outputs modeling wired as well as the wireless interconnects.

multiantenna interconnects, used for high-speed on-chip or chip-to-chip communication, such an assumption of having available high-resolution ADC and DAC components, cannot be made.

In case of on-chip multiconductor interconnects, the DAC and the ADC components are formed by the output or the input of a logic CMOS inverter, respectively. Hence, the ADC and DAC components, perform a *single-bit* conversion between the analog and the digital domain. With such coarse quantization, all state of the art techniques for MIMO signal processing fail.

In the case of wireless multiantenna interconnects for chip-to-chip communication the situation can be expected to be slightly better. However, because of the huge bandwidth, the requirements on conversion time are extremely high, such that only moderate resolution (4 –5 bits) ADC and DAC components are reasonable. As it turns out, such a moderately high resolution is still too low for reliable operation of state of the art MIMO signal processing.

In this section, we treat the ADC and DAC components as an integral part of the MIMO system. We develop signal processing and coding techniques, which utilize the information theoretic gains of MIMO systems with very-low to moderately-low resolution signal quantization. We first provide suitable design principles for low-latency channel-matched codes applied on general frequency selective MIMO channels, which are based on an information theoretic ground.

### **3.1. Single-bit ADC/DAC: Coding and performance limits**

Consider the MIMO channel with inter-symbol interference (ISI) and single-bit output quantization shown in Fig. 10. The channel has a memory of length *L* and it is governed by the channel law

$$\boldsymbol{r}\_{l} = \mathcal{Q}\left\{\boldsymbol{y}\_{l}\right\} = \mathcal{Q}\left\{\sum\_{k=0}^{L} \boldsymbol{H}\_{k} \boldsymbol{x}\_{l-k} + \eta\_{l}\right\}.\tag{1}$$

Here, <sup>H</sup>*<sup>k</sup>* <sup>∈</sup> **<sup>C</sup>***N*×*<sup>N</sup>* is the *<sup>k</sup>*-th channel matrix. <sup>x</sup>*<sup>k</sup>* ∈ X , <sup>η</sup>*<sup>k</sup>* <sup>∈</sup> **<sup>C</sup>***N*, <sup>y</sup>*<sup>k</sup>* <sup>∈</sup> **<sup>C</sup>***<sup>N</sup>* and <sup>r</sup>*<sup>k</sup>* ∈ Y <sup>=</sup> {*<sup>α</sup>* <sup>+</sup> <sup>j</sup>*β*|*α*, *<sup>β</sup>* ∈ {+1, <sup>−</sup>1}}*<sup>N</sup>* denote the channel input vector, the noise vector, the unquantized receive vector and the channel output vector, at the *k*-th time instant, respectively. The single-bit quantization operator Q returns the sign of the real and imaginary part of each component of the unquantized received signal r*t*, i.e.,

$$\mathcal{Q}\left\{y\_{l}\right\} = \text{sign}(\text{real}\{y\_{l}\}) + \mathbf{j} \cdot \text{sign}(\text{imag}\{y\_{l}\}).\tag{2}$$

The conditional probability of the channel output satisfies

$$\Pr(r\_t | x\_{-L'}^{\infty}, r\_1^{t-1}, r\_{t+1}^{\infty}) = \Pr(r\_t | x\_{t-L}^t), \ t \ge 0. \tag{3}$$

Here, x<sup>∞</sup> <sup>0</sup> and <sup>r</sup>*t*−<sup>1</sup> <sup>1</sup> stand for the sequences [x−*L*, *<sup>x</sup>*−*L*+1,...] and [r1, <sup>r</sup><sup>2</sup> ..., <sup>r</sup>*t*−1], respectively. The noise is additive white Gaussian with covariance matrix E[n*t*n<sup>H</sup> *<sup>t</sup>* ] = *<sup>σ</sup>*<sup>2</sup> *<sup>η</sup>* I*N*. The transmit signal energy is normalized to 1, that is �x*t*�<sup>2</sup> = 1. The signal-to-noise ratio is accordingly defined as

$$\text{SNR} = 1/\sigma\_{\eta}^{2}.\tag{4}$$

The channel transition probabilities can be calculated via

$$\Pr(r\_t|x\_{t-L}^t) = \prod\_{c \in \{\mathbb{R}, \mathbb{1}\}} \prod\_{i=1}^N \Phi\left(\frac{[y\_t]\_{c,i} \cdot [\bar{y}]\_{c,i}}{\sigma\_\eta / \sqrt{2}}\right) . \tag{5}$$

Here, [x*t*]**R**,*<sup>i</sup>* ([x*t*]**I**,*i*) denotes the *i*-th real (imaginary) component of the input vector x*t*, ¯y = ∑*L <sup>k</sup>*=<sup>0</sup> <sup>H</sup>*k*x*t*−*<sup>k</sup>* is the noise-free unquantized receive vector and <sup>Φ</sup>(*x*) = <sup>√</sup> 1 2*π x* <sup>−</sup><sup>∞</sup> *<sup>e</sup>*<sup>−</sup> *<sup>t</sup>*<sup>2</sup> <sup>2</sup> *dt* is the cumulative normal distribution function. The input symbols are modulated using QPSK (or BPSK with 1-bit DAC). Consequently, we consider ISI channels with the input and output cardinality |X | <sup>=</sup> |R| <sup>=</sup> <sup>4</sup>*N*.

### *3.1.1. Code-design*

10 Will-be-set-by-IN-TECH

η*t*

y*t* r*t*

*N*

x*<sup>t</sup>* x*t*−<sup>1</sup> x*t*−*<sup>L</sup>*

D

*N*

interconnects.

made.

ground.

by the channel law

H<sup>0</sup> H<sup>1</sup> H*<sup>L</sup>*

**Figure 10.** MIMO channel with ISI and single-bit outputs modeling wired as well as the wireless

multiantenna interconnects, used for high-speed on-chip or chip-to-chip communication, such an assumption of having available high-resolution ADC and DAC components, cannot be

In case of on-chip multiconductor interconnects, the DAC and the ADC components are formed by the output or the input of a logic CMOS inverter, respectively. Hence, the ADC and DAC components, perform a *single-bit* conversion between the analog and the digital domain. With such coarse quantization, all state of the art techniques for MIMO signal processing fail.

In the case of wireless multiantenna interconnects for chip-to-chip communication the situation can be expected to be slightly better. However, because of the huge bandwidth, the requirements on conversion time are extremely high, such that only moderate resolution (4 –5 bits) ADC and DAC components are reasonable. As it turns out, such a moderately high resolution is still too low for reliable operation of state of the art MIMO signal processing.

In this section, we treat the ADC and DAC components as an integral part of the MIMO system. We develop signal processing and coding techniques, which utilize the information theoretic gains of MIMO systems with very-low to moderately-low resolution signal quantization. We first provide suitable design principles for low-latency channel-matched codes applied on general frequency selective MIMO channels, which are based on an information theoretic

Consider the MIMO channel with inter-symbol interference (ISI) and single-bit output quantization shown in Fig. 10. The channel has a memory of length *L* and it is governed

Here, <sup>H</sup>*<sup>k</sup>* <sup>∈</sup> **<sup>C</sup>***N*×*<sup>N</sup>* is the *<sup>k</sup>*-th channel matrix. <sup>x</sup>*<sup>k</sup>* ∈ X , <sup>η</sup>*<sup>k</sup>* <sup>∈</sup> **<sup>C</sup>***N*, <sup>y</sup>*<sup>k</sup>* <sup>∈</sup> **<sup>C</sup>***<sup>N</sup>* and <sup>r</sup>*<sup>k</sup>* ∈ Y <sup>=</sup> {*<sup>α</sup>* <sup>+</sup> <sup>j</sup>*β*|*α*, *<sup>β</sup>* ∈ {+1, <sup>−</sup>1}}*<sup>N</sup>* denote the channel input vector, the noise vector, the unquantized receive vector and the channel output vector, at the *k*-th time instant, respectively. The single-bit quantization operator Q returns the sign of the real and imaginary part of each

*<sup>t</sup>*+1) = Pr(r*t*|x*<sup>t</sup>*

*<sup>k</sup>*=<sup>0</sup> <sup>H</sup>*k*x*t*−*<sup>k</sup>* + <sup>η</sup>*<sup>t</sup>*

Q {y*t*} = sign(real{y*t*}) + j · sign(imag{y*t*}). (2)

. (1)

*<sup>t</sup>*−*L*), *<sup>t</sup>* <sup>≥</sup> 0. (3)

 ∑*L*

**3.1. Single-bit ADC/DAC: Coding and performance limits**

r*<sup>t</sup>* = Q {y*t*} = Q

<sup>−</sup>*L*, <sup>r</sup>*t*−<sup>1</sup> <sup>1</sup> , <sup>r</sup><sup>∞</sup>

component of the unquantized received signal r*t*, i.e.,

The conditional probability of the channel output satisfies Pr(r*t*|x<sup>∞</sup>

We are looking for codes which maximally increase throughput and that allow fast decoding with good performance. In this way, the code design has to focus on finding low-complexity codes providing good coding gain while having low overhead both with respect to circuit complexity and power dissipation.

Even though linear block and convolution coding schemes are favorable candidates for error correction, they are not able to decrease power consumption and eliminate the residual error floor caused by the crosstalk even in the noiseless case [31]. This is due to their structural properties (linear codes) and the coarse quantization of the channel.

Therefore, the coding schemes which are needed are non-linear and – for having good performance and, at the same time, a low complexity – for instance have memory of order one. In [51–53], an information-theoretic framework was developed as a practical design guideline for novel codes. To this end, the following optimization has been considered in [52]

$$\mathcal{C}\_{M\_i}^{\text{uniform}} = \max\_{P\_{\vec{\eta}}(\cdot, \vec{\eta}) \in \mathcal{T}} \mathcal{Z}(\mathbf{z}; \mathbf{y}) \quad \text{s.t.} \quad P\_{\vec{\eta}} = \Pr(\mathcal{S}\_k = \boldsymbol{j} | \mathcal{S}\_{k-1} = \boldsymbol{i}) \in \{0, 2^{-K}\},$$

$$\mathcal{S}\_k = \mathbf{z}\_{k-M\_i+1}^k \in \mathcal{X}^{M\_i},$$

$$M\_{\vec{\sigma}} \ge L,\tag{6}$$

where *K* defines the rate *R* = *K*/(2*N*) of the code (for QPSK modulation) and *C* uniform *Ms* is the maximum channel capacity that can be attained with a homogeneous Markov source of order *Ms*. In general, the capacity of an unconstrained Markov source of order *Ms* [30] is higher than this *uniform capacity* (*Pi*,*<sup>j</sup>* takes 0 or 2−*K*), i.e., *C* uniform *Ms* ≤ *CMs*. This coding approach incorporates the following four ideas:

1. In order to avoid the complexity of maximizing an arbitrary Markov source, we restrict the optimization to homogeneous Markov sources.

	- 2. We choose the memory length *Ms* of the source to be roughly as long as the number of channel taps *L* but not shorter. The reason is that the information rate of a Markov source of order *Ms* = *L* is noticeably larger compared to the i.u.d. capacity, but memory lengths above *L* yield only a small additional gain in information rate.
	- 3. As we want to avoid the use of distribution shapers, the number of transmit symbols is fixed at 2*<sup>K</sup>* (irrespective of the current state). Thus, the encoder can be realized as a look-up table and we obtain the data encoding rule

$$x\_{\boldsymbol{\nu}} = \text{ENC}(d\_{\boldsymbol{\nu}} \left[ x\_{\boldsymbol{\nu}-1}, \dots, x\_{\boldsymbol{\nu}-M\_s} \right])\_{\boldsymbol{\nu}}$$

Channel Tx Rx

*xn*−<sup>1</sup> \ Data 000 001 010 011 100 101 110 111 0000 0001 0011 0111 1000 1100 1110 1111 0000 0001 0111 1000 1001 1100 1101 1111 0000 0010 0011 0100 1010 1011 1110 1111 0000 0010 0011 0100 1010 1011 1110 1111 0000 0001 0100 0101 0111 1100 1101 1111 0000 0001 0100 0101 0111 1100 1101 1111 0000 0011 0100 0110 0111 1100 1110 1111 0000 0001 0100 0110 0111 1100 1110 1111 0000 0001 0011 1000 1001 1011 1110 1111 0000 0001 0011 1000 1001 1011 1100 1111 0000 0010 0011 0100 1010 1011 1110 1111 0000 0010 0011 0100 1010 1011 1110 1111 0000 0001 0100 0101 0111 1100 1101 1111 0000 0001 0100 0101 0111 1100 1101 1111 0000 0010 0011 0110 0111 1000 1110 1111 0000 0001 0011 0111 1000 1100 1110 1111

At the receiver side, the decoder uses the value of the current and the previous channel

Obviously, the mapping done by this function performs a maximum-likelihood estimation of

Although this approach seems quite heuristic, its usefulness can be demonstrated by simulation. Table 1 lists the mapping function of a code designed for a bus with *N* = 4 mutually coupled, tapped *RC* lines as shown in Fig. 1 used at *T*CU = 9*RC*<sup>s</sup> = 1.5*RC*<sup>c</sup> symbol time, where *R* is the serial resistance, *C*s is the ground capacitance and *C*c is the coupling

<sup>1</sup> The reliability of the decoding can be improved if earlier or later outputs are considered using the BCJR algorithm

<sup>=</sup> arg max <sup>d</sup>

*n* as

d*<sup>n</sup>* = d

y*n*, y*n*−<sup>1</sup>

. (7)

Pr

d*<sup>n</sup>* dˆ

*<sup>N</sup>* <sup>x</sup>*<sup>n</sup>* <sup>r</sup>*<sup>n</sup> <sup>K</sup>*

DEC

Chip-to-Chip and On-Chip Communications 87

*n*

D

ENC

D

**Figure 12.** Noisy bus communication with a memory-based Code.

**Table 1.** The mapping function of a 3/4 optimized code: *x<sup>n</sup>* = ENC(*dn*,*xn*−1).

y*n*, y*n*−<sup>1</sup>

outputs to reconstruct an estimate of the data vector dˆ

1.

(forward-backward algorithm), at the cost of some latency.

dˆ *<sup>n</sup>* = *g* 

d*<sup>n</sup>* based on y*<sup>n</sup>* and y*n*−<sup>1</sup>

capacitance (c.f. section 2.1).

where <sup>d</sup>*<sup>n</sup>* ∈ {0, 1}*<sup>K</sup>* is the data vector of the source. Using QPSK modulation the realizable code rates *R* are

$$\frac{K}{2N} \in \left\{ \frac{1}{2N}, \frac{2}{2N}, \dots, 1 \right\}.$$

4. The optimized transition probabilities are uniformly distributed and at the same time they approximate the capacity-achieving input distribution. Hence, the optimized transition probability matrix *Pij* serves as an inner code that can be concatenated with an outer Turbo-like code, e.g. *low density parity check code* (c.f. section 4), in order to reach information rates (well) above the i.u.d. capacity [53].

Fig. 11 illustrates the channel model together with the encoder and decoder. The optimization

**Figure 11.** Encoding and Decoding for MIMO Channels with ISI and Single-Bit Output Quantization.

(6) is non-convex but can be solved by an efficient greedy algorithm [52] that delivers an optimized transition probability matrix P = [*Pi*,*j*] that maximizes the mutual information between the input and the output.

In Figure 12, a coded bus system employing a memory-based code with a code rate of *K*/*N* is shown. A fixed bus access time *T*cod CU is chosen such that two channel temporal taps are significant (*L* = 1). The encoding scheme is time-invariant and has the property that the data vector, d*<sup>n</sup>* = *d*1[*n*],..., *dK*[*n*] <sup>T</sup> <sup>∈</sup> 0, 1*K*×<sup>1</sup> , is encoded and decoded instantaneously (without latency). The actual code vector x*n* depends on the input data vector d*n* and the previous transmitted vector x*n*−1.

**Figure 12.** Noisy bus communication with a memory-based Code.

12 Will-be-set-by-IN-TECH

2. We choose the memory length *Ms* of the source to be roughly as long as the number of channel taps *L* but not shorter. The reason is that the information rate of a Markov source of order *Ms* = *L* is noticeably larger compared to the i.u.d. capacity, but memory lengths

3. As we want to avoid the use of distribution shapers, the number of transmit symbols is fixed at 2*<sup>K</sup>* (irrespective of the current state). Thus, the encoder can be realized as a look-up

<sup>x</sup>*<sup>n</sup>* = ENC(d*n*, [x*n*−1,...,x*n*−*Ms*]), where <sup>d</sup>*<sup>n</sup>* ∈ {0, 1}*<sup>K</sup>* is the data vector of the source. Using QPSK modulation the realizable

<sup>2</sup>*<sup>N</sup>* , ··· , 1

 .

 1 <sup>2</sup>*<sup>N</sup>* , <sup>2</sup>

4. The optimized transition probabilities are uniformly distributed and at the same time they approximate the capacity-achieving input distribution. Hence, the optimized transition probability matrix *Pij* serves as an inner code that can be concatenated with an outer Turbo-like code, e.g. *low density parity check code* (c.f. section 4), in order to reach

Fig. 11 illustrates the channel model together with the encoder and decoder. The optimization

H<sup>0</sup> H<sup>1</sup> H*<sup>L</sup>*

**Figure 11.** Encoding and Decoding for MIMO Channels with ISI and Single-Bit Output Quantization. (6) is non-convex but can be solved by an efficient greedy algorithm [52] that delivers an optimized transition probability matrix P = [*Pi*,*j*] that maximizes the mutual information

In Figure 12, a coded bus system employing a memory-based code with a code rate of *K*/*N*

significant (*L* = 1). The encoding scheme is time-invariant and has the property that the

0, 1*K*×<sup>1</sup>

(without latency). The actual code vector x*n* depends on the input data vector d*n* and the

<sup>T</sup> <sup>∈</sup>

x*n*−<sup>1</sup> x*n*−*<sup>L</sup>*

η*n*

r*n* y*n*

CU is chosen such that two channel temporal taps are

, is encoded and decoded instantaneously

dˆ *n*

DEC

above *L* yield only a small additional gain in information rate.

*K* 2*N* ∈

information rates (well) above the i.u.d. capacity [53].

T

x*n*−<sup>1</sup>

ENC

x*n*−*Ms*

data vector, d*<sup>n</sup>* =

d*n*

x*n*

T

between the input and the output.

previous transmitted vector x*n*−1.

is shown. A fixed bus access time *T*cod

T

Channel

*d*1[*n*],..., *dK*[*n*]

table and we obtain the data encoding rule

code rates *R* are


**Table 1.** The mapping function of a 3/4 optimized code: *x<sup>n</sup>* = ENC(*dn*,*xn*−1).

At the receiver side, the decoder uses the value of the current and the previous channel outputs to reconstruct an estimate of the data vector dˆ *n* as

$$\hat{\mathbf{d}}\_{\boldsymbol{n}} = \mathcal{g}(y\_{\boldsymbol{n}\prime} y\_{\boldsymbol{n}-1}) = \arg\max\_{\mathbf{d}} \Pr\left(\mathbf{d}\_{\boldsymbol{n}} = \mathbf{d} \mid y\_{\boldsymbol{n}\prime} y\_{\boldsymbol{n}-1}\right). \tag{7}$$

Obviously, the mapping done by this function performs a maximum-likelihood estimation of d*<sup>n</sup>* based on y*<sup>n</sup>* and y*n*−<sup>1</sup> 1.

Although this approach seems quite heuristic, its usefulness can be demonstrated by simulation. Table 1 lists the mapping function of a code designed for a bus with *N* = 4 mutually coupled, tapped *RC* lines as shown in Fig. 1 used at *T*CU = 9*RC*<sup>s</sup> = 1.5*RC*<sup>c</sup> symbol time, where *R* is the serial resistance, *C*s is the ground capacitance and *C*c is the coupling capacitance (c.f. section 2.1).

<sup>1</sup> The reliability of the decoding can be improved if earlier or later outputs are considered using the BCJR algorithm (forward-backward algorithm), at the cost of some latency.

#### 14 Will-be-set-by-IN-TECH 88 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Chip-to-Chip and On-Chip Communications <sup>15</sup>

Its performance in terms of symbol error rate (SER) when applied to a noisy bus system, compared to uncoded transmission, is shown in Figure 13. The uncoded transmission reveals an error floor (a residual SER at vanishing noise variance) due to signaling belong the *RC*–specific time. However, as we see in Figure 13, the optimized code does not see any error floor. Besides, it turns out that the achievable power savings of this code (in terms of energy per transmitted information bit) is 40%, without taking into account the power overhead of the codec circuit. The SER curve of a space-only code, which has been optimized by exhaustive search, is also plotted. Due to its simplicity, this code performs inherently worse than the discussed memory-based code. Although several coding schemes can be found in the literature [29, 32], such a unified framework that jointly address power, rate, and reliability aspects, simultaneously is new.

studied theoretically and experimentally. Thereby, perfect channel state information (CSI) at the receiver is assumed, which can be obtained even with coarse quantization as discussed in

In [33], the joint optimization of the linear receiver and the quantizer in a MIMO system is addressed. The figure of merit that has been used for the design of the optimum quantizer and receiver is the *mean square error* (MSE). Based on this MSE approach, the communication performance (in terms of channel capacity) of the quantized MIMO channel is studied. Our work [34] generalizes this modified MMSE filter to frequency selective channels. Motivated by the same approach, the authors of [36] optimized the Decision Feedback Equalizer (DFE) for

In this and the following Section, we provide a summary of these works. Throughout

Let us now consider a point to point MIMO Gaussian channel, where the transmitter operates *M* antennas and the receiver employs *N* antennas. Figure 14 shows the general form of a quantized MIMO system, where <sup>H</sup> <sup>∈</sup> **<sup>C</sup>***N*×*<sup>M</sup>* is the channel matrix. For simplicity, inter-symbol interference (ISI) is ignored, even though considering it would be straightforward. The vector <sup>x</sup> <sup>∈</sup> **<sup>C</sup>***<sup>M</sup>* comprises the *<sup>M</sup>* transmitted symbols with zero-mean and covariance R*xx* = E[xxH]. The vector η refers to zero-mean complex circularly symmetric Gaussian noise with a covariance matrix <sup>R</sup>*ηη* <sup>=</sup> <sup>E</sup>[ηηH], while <sup>y</sup> <sup>∈</sup> **<sup>C</sup>***<sup>N</sup>* is the unquantized channel

In our system, the real parts *yi*,R and the imaginary parts *yi*,I of the receive signals *yi*, 1 ≤ *i* ≤ *N*, are each quantized by a *b*-bit resolution uniform/non-uniform scalar quantizer. Thus, the

where *Q*(·) denotes the quantization operation and *qi*,*<sup>l</sup>* is the resulting quantization error.

Our aim is to choose the quantizer and the receive matrix G minimizing the MSE =

<sup>2</sup>], taking into account the quantization effect. Since the ADC can drastically affect

*i*,*l* ] *ryi*,*lyi*,*<sup>l</sup>*

R, I

Hermitian transpose, complex conjugate, and trace of a matrix, respectively.

*ri*,*<sup>l</sup>* <sup>=</sup> *<sup>Q</sup>*(*yi*,*l*) = *yi*,*<sup>l</sup>* <sup>+</sup> *qi*,*l*, *<sup>l</sup>* <sup>∈</sup>

the performance of the system, it should be also designed carefully.

Each quantization process can be given a distortion factor *ρ*

of quantization noise generated, which is defined as follows

The matrix <sup>G</sup> <sup>∈</sup> **<sup>C</sup>***M*×*<sup>N</sup>* represents the receive filter, which delivers the estimate ˆ<sup>x</sup>

*ρ* (*i*,*l*) *<sup>q</sup>* <sup>=</sup> <sup>E</sup>[*q*<sup>2</sup>

]. The operators . (•)T, (•)H, (•)∗, tr[•] stand for transpose,

y = Hx + η. (8)

xˆ = Gr. (10)

(*i*,*l*)

, 1 ≤ *i* ≤ *N*, (9)

Chip-to-Chip and On-Chip Communications 89

*<sup>q</sup>* to indicate the relative amount

, (11)

Section 3.3.

the flat MIMO channel with quantized outputs.

these sections, *rαβ* denotes E[*αβ*<sup>∗</sup>

resulting quantized signals are given by

*3.2.1. System model*

output:

<sup>E</sup>[�x<sup>ˆ</sup> <sup>−</sup>x�<sup>2</sup>

*3.2.2. Quantizer characterization*

**Figure 13.** Symbol error rate (SER) as function of *V*<sup>2</sup> DD/*V*<sup>2</sup> <sup>N</sup> for an interconnect with four lines, employing a memory-based code of rate 3/4, compared to the uncoded case. The performance of a space-only code is also plotted.

We note that, for large buses, it is impractical to encode all bits at once because of the large complexity in the design and the implementation of the codec circuit. Therefore, partial coding can be employed in which the bus is partitioned into sub-buses of smaller width, which are encoded separately. The partitioning requires some additional wires since a shielding wire has to be placed between every two adjacent sub-buses.

### **3.2. Low-resolution ADC: Linear signal-processing**

In the following, we concentrated on receive signal processing and our aim is to study the applicability of standard equalization techniques for our application, where the receiver is equipped with a low to moderate ADC for each antenna or port. A modified version of the standard linear receiver designs is presented in the context of MIMO communication with quantized output, taking into account the presence of the quantizer. An essential aspect of our analysis is that no assumption of uncorrelated white quantization errors is made. The performance of the modified receiver designs as well as the effects of quantization are studied theoretically and experimentally. Thereby, perfect channel state information (CSI) at the receiver is assumed, which can be obtained even with coarse quantization as discussed in Section 3.3.

In [33], the joint optimization of the linear receiver and the quantizer in a MIMO system is addressed. The figure of merit that has been used for the design of the optimum quantizer and receiver is the *mean square error* (MSE). Based on this MSE approach, the communication performance (in terms of channel capacity) of the quantized MIMO channel is studied. Our work [34] generalizes this modified MMSE filter to frequency selective channels. Motivated by the same approach, the authors of [36] optimized the Decision Feedback Equalizer (DFE) for the flat MIMO channel with quantized outputs.

In this and the following Section, we provide a summary of these works. Throughout these sections, *rαβ* denotes E[*αβ*<sup>∗</sup> ]. The operators . (•)T, (•)H, (•)∗, tr[•] stand for transpose, Hermitian transpose, complex conjugate, and trace of a matrix, respectively.

### *3.2.1. System model*

14 Will-be-set-by-IN-TECH

Its performance in terms of symbol error rate (SER) when applied to a noisy bus system, compared to uncoded transmission, is shown in Figure 13. The uncoded transmission reveals an error floor (a residual SER at vanishing noise variance) due to signaling belong the *RC*–specific time. However, as we see in Figure 13, the optimized code does not see any error floor. Besides, it turns out that the achievable power savings of this code (in terms of energy per transmitted information bit) is 40%, without taking into account the power overhead of the codec circuit. The SER curve of a space-only code, which has been optimized by exhaustive search, is also plotted. Due to its simplicity, this code performs inherently worse than the discussed memory-based code. Although several coding schemes can be found in the literature [29, 32], such a unified framework that jointly address power, rate, and reliability

0 5 10 15 20

DD/*V*<sup>2</sup>

<sup>N</sup> for an interconnect with four lines,

*V*2 DD/*V*<sup>2</sup> N

employing a memory-based code of rate 3/4, compared to the uncoded case. The performance of a

We note that, for large buses, it is impractical to encode all bits at once because of the large complexity in the design and the implementation of the codec circuit. Therefore, partial coding can be employed in which the bus is partitioned into sub-buses of smaller width, which are encoded separately. The partitioning requires some additional wires since a shielding wire

In the following, we concentrated on receive signal processing and our aim is to study the applicability of standard equalization techniques for our application, where the receiver is equipped with a low to moderate ADC for each antenna or port. A modified version of the standard linear receiver designs is presented in the context of MIMO communication with quantized output, taking into account the presence of the quantizer. An essential aspect of our analysis is that no assumption of uncorrelated white quantization errors is made. The performance of the modified receiver designs as well as the effects of quantization are

Uncoded Space−only Code Memory based Code

aspects, simultaneously is new.

10−3

**Figure 13.** Symbol error rate (SER) as function of *V*<sup>2</sup>

has to be placed between every two adjacent sub-buses.

**3.2. Low-resolution ADC: Linear signal-processing**

space-only code is also plotted.

10−2

SER

10−1

100

Let us now consider a point to point MIMO Gaussian channel, where the transmitter operates *M* antennas and the receiver employs *N* antennas. Figure 14 shows the general form of a quantized MIMO system, where <sup>H</sup> <sup>∈</sup> **<sup>C</sup>***N*×*<sup>M</sup>* is the channel matrix. For simplicity, inter-symbol interference (ISI) is ignored, even though considering it would be straightforward. The vector <sup>x</sup> <sup>∈</sup> **<sup>C</sup>***<sup>M</sup>* comprises the *<sup>M</sup>* transmitted symbols with zero-mean and covariance R*xx* = E[xxH]. The vector η refers to zero-mean complex circularly symmetric Gaussian noise with a covariance matrix <sup>R</sup>*ηη* <sup>=</sup> <sup>E</sup>[ηηH], while <sup>y</sup> <sup>∈</sup> **<sup>C</sup>***<sup>N</sup>* is the unquantized channel output:

$$y = Hx + \eta.\tag{8}$$

In our system, the real parts *yi*,R and the imaginary parts *yi*,I of the receive signals *yi*, 1 ≤ *i* ≤ *N*, are each quantized by a *b*-bit resolution uniform/non-uniform scalar quantizer. Thus, the resulting quantized signals are given by

$$r\_{i,l} = \mathbb{Q}(y\_{i,l}) = y\_{i,l} + q\_{i,l}, \; l \in \{\mathbb{R}, \mathbf{I}\}, \; 1 \le i \le N,\tag{9}$$

where *Q*(·) denotes the quantization operation and *qi*,*<sup>l</sup>* is the resulting quantization error. The matrix <sup>G</sup> <sup>∈</sup> **<sup>C</sup>***M*×*<sup>N</sup>* represents the receive filter, which delivers the estimate ˆ<sup>x</sup>

$$
\hat{x} = Gr.\tag{10}
$$

Our aim is to choose the quantizer and the receive matrix G minimizing the MSE = <sup>E</sup>[�x<sup>ˆ</sup> <sup>−</sup>x�<sup>2</sup> <sup>2</sup>], taking into account the quantization effect. Since the ADC can drastically affect the performance of the system, it should be also designed carefully.

### *3.2.2. Quantizer characterization*

Each quantization process can be given a distortion factor *ρ* (*i*,*l*) *<sup>q</sup>* to indicate the relative amount of quantization noise generated, which is defined as follows

$$\rho\_q^{(i,l)} = \frac{\mathbb{E}[q\_{i,l}^2]}{r\_{y\_{i\ell}y\_{i\ell}}},\tag{11}$$

### **Figure 14.** Quantized MIMO System.

where *ryi*,*lyi*,*<sup>l</sup>* = E[*y*<sup>2</sup> *i*,*l* ] is the variance of *yi*,*<sup>l</sup>* and the distortion factor *ρ* (*i*,*l*) *<sup>q</sup>* depends on the number of quantization bits *b*, the quantizer type (uniform or non-uniform) and the probability density function of *yi*,*l*. Note that the signal-to-quantization noise ratio (SQNR) has an inverse relationship with regard to the distortion factor. The uniform/non-uniform quantizer design is based on minimizing the *mean square error* (distortion) between the input *yi*,*<sup>l</sup>* and the output *ri*,*<sup>l</sup>* of each quantizer. In other words, the SQNR values are maximized. With this optimal design of the scalar finite resolution quantizer, whether uniform or not, the following equations hold for all 0 ≤ *i* ≤ *N*, *l* ∈ {*R*, *I*} [35, 37, 38]

$$\mathbb{E}[r\_{i,l}q\_{i,l}] = 0\tag{12}$$

and R*rr* can be expressed as

*ryiyj r*−<sup>1</sup> *yjyj* R*rr* = E[(y + q)(y + q)H] = R*yy* + R*yq* + R<sup>H</sup>

from all other random variables of the system. First we calculate *ryiqj* = E[*yiq*<sup>∗</sup>

*<sup>j</sup>* ] = E*yj* E[*yiq*<sup>∗</sup> *<sup>j</sup>* |*yj*] 

> = E*yj*

≈ E*yj ryiyj r*−<sup>1</sup> *yjyj yj*E[*q*<sup>∗</sup> *<sup>j</sup>* |*yj*] 

= *ryiyj*

= −*ρqryiyj*

where nondiag(A) obtained from a matrix A by setting its diagonal elements to zero.

In summary, we get from (23) and (24) the following expression for the Wiener filter from (15)

<sup>R</sup>*xx*−(1−*ρq*)R*xy*(R*yy*−*ρq*nondiag(R*yy*))−<sup>1</sup>

*r*−<sup>1</sup> *yjyj* E[*yjq*<sup>∗</sup> *j* ]

Note that, in (19), we approximate the Bayesian estimator E[*yi*|*yj*] with the linear estimator

*yj*, which holds with equality if the vector y is jointly Gaussian distributed. Eq. (20)

E[*yiq*<sup>∗</sup>

follows from (14). Summarizing the results of (14) and (20), we obtain

Similarly, we evaluate *rqiqj* for *i* �= *j* using (21), and with (14) we arrive at

Also in a similar way, we get <sup>R</sup>*xq* <sup>=</sup> <sup>E</sup>[xqH] ≈ −*ρq*R*xy*, and (17) becomes

Inserting the expressions (21) and (22) into (18), we obtain

operating on quantized data

and for the resulting MSE, we obtain using (16)

MSEWFQ <sup>≈</sup>tr

We obtain R*yy* and R*xy* easily from our system model

We have to determine the linear filter G as a function of the channel parameters and the quantization distortion factor *ρq*. To this end, we derive all needed covariance matrices by using the fact that the quantization error *qi*, conditioned on *yi*, is statistically independent

E[*yi*|*yj*]E[*q*<sup>∗</sup>

*<sup>j</sup>* |*yj*]  *yq* + R*qq*. (18)

Chip-to-Chip and On-Chip Communications 91

. (20)

R*yq* ≈ −*ρq*R*yy*. (21)

R*qq* ≈ *ρq*R*yy* − (1 − *ρq*)*ρq*nondiag(R*yy*), (22)

R*rr* ≈ (1 − *ρq*)(R*yy* − *ρq*nondiag(R*yy*)). (23)

<sup>G</sup>WFQ <sup>≈</sup> <sup>R</sup>*xy*(R*yy* <sup>−</sup> *<sup>ρ</sup>q*nondiag(R*yy*))<sup>−</sup>1, (25)

R*xr* ≈ (1 − *ρq*)R*xy*. (24)

R*yy* = R*ηη* + HR*xx*HH, (27)

R*xy* = R*xx*HH. (28)

R<sup>H</sup> *xy*

. (26)

*<sup>j</sup>* ] for *i* �= *j*

(19)

$$\mathbb{E}[y\_{i,l}\eta\_{i,l}] = -\rho\_q^{(i,l)}r\_{y\_{i,l}y\_{i,l}}.\tag{13}$$

Obviously, (13) follows from (11) and (12). Under multipath propagation conditions and for large number of antennas, the quantizer input signals *yi*,*<sup>l</sup>* will be approximately Gaussian distributed and thus, they undergo nearly the same distortion factor *ρq*, i.e., *ρ* (*i*,*l*) *<sup>q</sup>* = *ρ<sup>q</sup>* ∀*i*∀*l*. Furthermore, the optimal parameters of the uniform as well as the non-uniform quantizer and the resulting distortion factor *ρq* for Gaussian distributed signal are tabulated in [35] for different resolutions *b*.

Now, let *qi* = *qi*,R + j*qi*,I be the complex quantization error. Under the assumption of uncorrelated real and imaginary part of *yi*, the following relations are obtained

$$r\_{q\_i q\_l} = \mathcal{E}[q\_i q\_i^\*] = \rho\_q r\_{y\_i y\_{i'}} \text{ and } r\_{y\_i q\_l} = \mathcal{E}[y\_i q\_i^\*] = -\rho\_q r\_{y\_i y\_l}.\tag{14}$$

This particular choice of the (non-)uniform scalar quantizer minimizing the distortion between r and y, combined with the receiver developed in the next Section, is also optimal with respect to the total MSE between the transmitted symbol vector x and the estimated symbol vector ˆx, as we will see later.

### *3.2.3. Nearly optimal linear receiver*

The linear receiver <sup>G</sup> that minimizes the MSE, E[�ε�<sup>2</sup> <sup>2</sup>] = <sup>E</sup>[�<sup>x</sup> <sup>−</sup> <sup>x</sup><sup>ˆ</sup> �<sup>2</sup> <sup>2</sup>] = <sup>E</sup>[�<sup>x</sup> <sup>−</sup> Gr�<sup>2</sup> <sup>2</sup>], can be written as:

$$\mathbf{G} = \mathbf{R}\_{\text{AT}} \mathbf{R}\_{\text{rr}}^{-1} \tag{15}$$

and the resulting MSE equals

$$\text{MSE} = \text{tr}\left(\mathbf{R}\_{\varepsilon\varepsilon}\right) = \text{tr}\left(\mathbf{R}\_{\text{xx}} - \mathbf{R}\_{\text{x}\tau}\mathbf{R}\_{\tau\tau}^{-1}\mathbf{R}\_{\text{x}\tau}^{\text{H}}\right),\tag{16}$$

where R*xr* equals

$$\mathbf{R\_{xr}} = \mathbb{E}[x r^{\mathbf{H}}] = \mathbb{E}[x(y+\mathbf{q})^{\mathbf{H}}] = \mathbf{R\_{xy}} + \mathbf{R\_{xq}} \tag{17}$$

#### 90 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Chip-to-Chip and On-Chip Communications <sup>17</sup> Chip-to-Chip and On-Chip Communications 91

and R*rr* can be expressed as

16 Will-be-set-by-IN-TECH

H G x xˆ

] is the variance of *yi*,*<sup>l</sup>* and the distortion factor *ρ*

(*i*,*l*)

E[*ri*,*lqi*,*l*] = 0 (12)

*<sup>i</sup>* ] = −*ρqryiyi*

<sup>2</sup>] = <sup>E</sup>[�<sup>x</sup> <sup>−</sup> <sup>x</sup><sup>ˆ</sup> �<sup>2</sup>

*rr* <sup>R</sup><sup>H</sup> *xr* 

R*xr* = E[xrH] = E[x(y + q)H] = R*xy* + R*xq*, (17)

<sup>G</sup> = <sup>R</sup>*xr*R−<sup>1</sup> *rr* , (15)

*<sup>q</sup> ryi*,*<sup>l</sup> yi*,*<sup>l</sup>* . (13)

the number of quantization bits *b*, the quantizer type (uniform or non-uniform) and the probability density function of *yi*,*l*. Note that the signal-to-quantization noise ratio (SQNR) has an inverse relationship with regard to the distortion factor. The uniform/non-uniform quantizer design is based on minimizing the *mean square error* (distortion) between the input *yi*,*<sup>l</sup>* and the output *ri*,*<sup>l</sup>* of each quantizer. In other words, the SQNR values are maximized. With this optimal design of the scalar finite resolution quantizer, whether uniform or not, the

E[*yi*,*lqi*,*l*] = −*ρ*

distributed and thus, they undergo nearly the same distortion factor *ρq*, i.e., *ρ*

uncorrelated real and imaginary part of *yi*, the following relations are obtained

*<sup>i</sup>* ] = *ρqryiyi*

MSE = tr(R*εε*) = tr

*rqiqi* = E[*qiq*<sup>∗</sup>

The linear receiver <sup>G</sup> that minimizes the MSE, E[�ε�<sup>2</sup>

symbol vector ˆx, as we will see later.

*3.2.3. Nearly optimal linear receiver*

and the resulting MSE equals

be written as:

where R*xr* equals

Obviously, (13) follows from (11) and (12). Under multipath propagation conditions and for large number of antennas, the quantizer input signals *yi*,*<sup>l</sup>* will be approximately Gaussian

Furthermore, the optimal parameters of the uniform as well as the non-uniform quantizer and the resulting distortion factor *ρq* for Gaussian distributed signal are tabulated in [35] for

Now, let *qi* = *qi*,R + j*qi*,I be the complex quantization error. Under the assumption of

This particular choice of the (non-)uniform scalar quantizer minimizing the distortion between r and y, combined with the receiver developed in the next Section, is also optimal with respect to the total MSE between the transmitted symbol vector x and the estimated

<sup>R</sup>*xx* <sup>−</sup> <sup>R</sup>*xr*R−<sup>1</sup>

, and *ryiqi* = E[*yiq*<sup>∗</sup>

(*i*,*l*)

(*i*,*l*)

. (14)

<sup>2</sup>] = <sup>E</sup>[�<sup>x</sup> <sup>−</sup> Gr�<sup>2</sup>

, (16)

<sup>2</sup>], can

*<sup>q</sup>* = *ρ<sup>q</sup>* ∀*i*∀*l*.

*<sup>q</sup>* depends on

η

*M N*

following equations hold for all 0 ≤ *i* ≤ *N*, *l* ∈ {*R*, *I*} [35, 37, 38]

**Figure 14.** Quantized MIMO System.

*i*,*l*

where *ryi*,*lyi*,*<sup>l</sup>* = E[*y*<sup>2</sup>

different resolutions *b*.

y r *Q*(•)

$$\mathbf{R}\_{\mathbf{J}\dagger} = \mathbf{E}[(\mathbf{y} + \mathbf{q})(\mathbf{y} + \mathbf{q})^{\mathbf{H}}] = \mathbf{R}\_{\mathbf{y}\mathbf{y}} + \mathbf{R}\_{\mathbf{y}\mathbf{q}} + \mathbf{R}\_{\mathbf{y}\mathbf{q}}^{\mathbf{H}} + \mathbf{R}\_{\mathbf{q}\mathbf{q}}.\tag{18}$$

We have to determine the linear filter G as a function of the channel parameters and the quantization distortion factor *ρq*. To this end, we derive all needed covariance matrices by using the fact that the quantization error *qi*, conditioned on *yi*, is statistically independent from all other random variables of the system. First we calculate *ryiqj* = E[*yiq*<sup>∗</sup> *<sup>j</sup>* ] for *i* �= *j*

$$\begin{aligned} \mathbb{E}[y\_i q\_j^\*] &= \mathbb{E}\_{y\_j} [\mathbb{E}[y\_i q\_j^\* | y\_j]] \\ &= \mathbb{E}\_{y\_j} [\mathbb{E}[y\_i | y\_j] \mathbb{E}[q\_j^\* | y\_j]] \\ &\approx \mathbb{E}\_{y\_j} [r\_{y\_j y\_j} r\_{y\_j y\_j}^{-1} y\_j \mathbb{E}[q\_j^\* | y\_j]] \\ &= r\_{y\_j y\_j} r\_{y\_j y\_j}^{-1} \mathbb{E}[y\_j q\_j^\*] \\ &= -\rho\_l r\_{y\_j y\_j}. \tag{20} \end{aligned} \tag{20}$$

Note that, in (19), we approximate the Bayesian estimator E[*yi*|*yj*] with the linear estimator *ryiyj r*−<sup>1</sup> *yjyj yj*, which holds with equality if the vector y is jointly Gaussian distributed. Eq. (20) follows from (14). Summarizing the results of (14) and (20), we obtain

$$\mathbf{R}\_{yq} \approx -\rho\_q \mathbf{R}\_{yy}.\tag{21}$$

Similarly, we evaluate *rqiqj* for *i* �= *j* using (21), and with (14) we arrive at

$$\mathbf{R}\_{qq} \approx \rho\_q \mathbf{R}\_{yy} - (1 - \rho\_q)\rho\_q \mathbf{n} \text{condiag}(\mathbf{R}\_{yy}),\tag{22}$$

where nondiag(A) obtained from a matrix A by setting its diagonal elements to zero. Inserting the expressions (21) and (22) into (18), we obtain

$$\mathbf{R}\_{rr} \approx (1 - \rho\_q)(\mathbf{R}\_{yy} - \rho\_q \mathbf{n} \text{condia}(\mathbf{R}\_{yy})).\tag{23}$$

Also in a similar way, we get <sup>R</sup>*xq* <sup>=</sup> <sup>E</sup>[xqH] ≈ −*ρq*R*xy*, and (17) becomes

$$\mathbf{R}\_{\mathbf{x}r} \approx (1 - \rho\_q)\mathbf{R}\_{\mathbf{x}y}.\tag{24}$$

In summary, we get from (23) and (24) the following expression for the Wiener filter from (15) operating on quantized data

$$\mathbf{G\_{WFQ}} \approx \mathbf{R\_{xy}} (\mathbf{R\_{yy}} - \rho\_{\eta} \mathbf{nondig}(\mathbf{R\_{yy}}))^{-1},\tag{25}$$

and for the resulting MSE, we obtain using (16)

$$\text{MSE}\_{\text{WFQ}} \approx \text{tr}\left[\mathbf{R}\_{\text{xx}} - (1 - \rho\_{\eta})\mathbf{R}\_{\text{xy}}(\mathbf{R}\_{\text{yy}} - \rho\_{\eta}\text{nonddiag}(\mathbf{R}\_{\text{yy}}))^{-1}\mathbf{R}\_{\text{xy}}^{\text{H}}\right].\tag{26}$$

We obtain R*yy* and R*xy* easily from our system model

$$R\_{\mathcal{Y}\mathcal{Y}} = R\_{\eta\eta} + HR\_{\text{xx}}H^{\text{H}},\tag{27}$$

$$R\_{xy} = R\_{xx}H^{\text{H}}.\tag{28}$$

#### 18 Will-be-set-by-IN-TECH 92 Ultra-Wideband Radio Technologies for Communications, Localization and Sensor Applications Chip-to-Chip and On-Chip Communications <sup>19</sup>

Let us examine the first derivative of the MSEWFQ in (26) with respect to *ρ<sup>q</sup>*

$$\frac{\partial \text{MSE}\_{\text{WFQ}}}{\partial \rho\_q} = \text{tr}\left[\mathbf{G}\_{\text{WFQ}} \text{diag}(\mathbf{R}\_{\mathcal{Y}\mathcal{Y}}) \mathbf{G}\_{\text{WFQ}}^{\text{H}}\right] > 0,\tag{29}$$

information becomes

*3.2.5. Simulation results*

that is, R*xx* = *σ*<sup>2</sup>

lim SNR→0

showed that the above approximation is asymptotically exact.

*<sup>x</sup>* **I** and R*ηη* = *σ*<sup>2</sup>

case when no quantization is applied.

approach, which hampers the analysis of performance.

**3.3. Channel estimation**

*I*(x, r) *I*(x, y)

In other words, the power penalty due to the 1-bit quantization is approximately equal *<sup>π</sup>*

dB) at low SNR. This shows that mono-bit ADCs may be used to save system power without an excessive degradation in performance, and confirms the significant potential of the coarsely quantized UWB MIMO channel. Using a different approach, [40] presented a similar result, and

The performance of the modified Wiener filter for a 4- and 5-bit quantized output MIMO system (WFQ), in terms of BER averaged over 1000 channel realizations, is shown in Figure 15 for a 10×10 MIMO system (QPSK), compared with the conventional Wiener filter (WF) and Zero-forcing filter (ZF). The symbols and the noise samples are assumed to be uncorrelated,

SNR <sup>=</sup> <sup>10</sup> · log10

Furthermore, we used a generic channel model, where the entries of H are complex-valued realizations of independent zero-mean Gaussian random variables with unit variance. Clearly, the modified Wiener filter outperforms the conventional Wiener filter at high SNR. This is because the effect of the quantization error is more pronounced at higher SNR values when compared to the additive Gaussian noise variance. Since the conventional Wiener filter converges to the ZF-filter at high SNR values and loses its regularized structure, its performance degrades asymptotically to the performance of the ZF-filter, when operating on quantized data. For comparison, we also plot the BER curves for the WF and ZF filter, for the

Because in general, the MIMO channel cannot be assumed known a-priori, a channel estimation has to be performed. In practice, it is highly desirable that the channel is estimated directly by the communication device – in our case by on-chip digital circuitry. However, this implies that the channel estimator is restricted to use received signal samples of a pilot sequence after single-bit quantization in the extreme case. This motivates investigation of channel estimation with coarse quantization. This problem was first addressed by [27], where a maximum likelihood (ML) channel estimation with quantized observation is presented. In general, the solution cannot be given in closed form, but requires an iterative numerical

In [28], it has been shown that – in contrast to unquantized channel estimation – different orthogonal pilot sequences (with same average total transmit power and same length) yield different performances. Especially, orthogonality in the time-domain (time-multiplexed pilots) can be preferable to orthogonality in space. With orthogonal pilots

*<sup>η</sup>* **I**. Hereby, the SNR (in dB) is defined as

*σ*2 *x σ*2 *η*  *b*=1

≈ 2

*<sup>π</sup>*. (35)

Chip-to-Chip and On-Chip Communications 93

. (36)

<sup>2</sup> (1.96

where GWFQ is given in (25). Therefore the MSEWFQ is monotonically increasing in *ρq*. Since we choose the quantizer to minimize the distortion factor *ρq*, our receiver and quantizer designs are jointly optimum with respect to the total MSE.

### *3.2.4. Lower bound on the mutual information and the capacity*

In this section, we develop a lower bound on the mutual information rate between the input sequence x and the quantized output sequence r of the system in Figure 14, based on our MSE approach. Generally, the mutual information of this channel can be expressed as [26]

$$I(x,r) = H(x) - H(x|r). \tag{30}$$

Given R*xx* under a power constraint tr(R*xx*) ≤ *P*Tr, we choose x to be Gaussian, which is not necessarily the capacity achieving distribution for our quantized system. Then, we can obtain a lower bound for *I*(x, r) (in bit/transmission) as

$$I(x,r) = \log\_2 |\mathcal{R}\_{\text{xx}}| - h(x|r)$$

$$= \log\_2 |\mathcal{R}\_{\text{xx}}| - h(x - \hat{x}|r)$$

$$\ge \log\_2 |\mathcal{R}\_{\text{xx}}| - h(x - \hat{x}) \tag{31}$$

$$\geq \log\_2 \frac{|\mathcal{R}\_{\text{xx}}|}{|\mathcal{R}\_{\text{ee}}|}. \tag{32}$$

Since conditioning reduces entropy, we obtain inequality (31). On the other hand, the second term in (31) is upper bounded by the entropy of a Gaussian random variable whose covariance is equal to the error covariance matrix R*εε* of the linear MMSE estimate of x. Finally, we get using (26) and (28)

$$I(x,r) \overset{>}{\approx} -\log\_2\left|\mathbf{I} - (1-\rho\_\theta)\mathbf{R}\_{xy}(\mathbf{R}\_{yy} - \rho\_\theta \mathbf{n} \text{condian}(\mathbf{R}\_{yy}))^{-1}\mathbf{H}\right|.\tag{33}$$

Considering the case of low SNR values, we get easily with R*yy* ≈ R*ηη*, (33) and (28), the following first order approximation of the mutual information2 3

$$I(x, r) \overset{>}{\underset{\approx}{\approx}} (1 - \rho\_q) \text{tr}[\mathbf{R}\_{\text{xx}} \mathbf{H}^\text{H} \mathbf{R}\_{\eta\eta}^{-1} \mathbf{H}] / \log(2). \tag{34}$$

Compared with the mutual information *I*(x,y) for the unquantized case, also at low SNR [39], the mutual information for the quantized channel degrades only by the factor (1 − *ρq*). For the spacial case *<sup>b</sup>* <sup>=</sup> 1, we have *<sup>ρ</sup>q*|*b*=<sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>2</sup> *<sup>π</sup>* (see [35]) and the degradation of the mutual

<sup>2</sup> We assume also that *<sup>ρ</sup><sup>q</sup>* � 1 (or <sup>R</sup>*ηη* is diagonal). <sup>3</sup> Note that log <sup>|</sup>**<sup>I</sup>** <sup>+</sup> <sup>Δ</sup>X| ≈ tr(ΔX).

information becomes

18 Will-be-set-by-IN-TECH

where GWFQ is given in (25). Therefore the MSEWFQ is monotonically increasing in *ρq*. Since we choose the quantizer to minimize the distortion factor *ρq*, our receiver and quantizer

In this section, we develop a lower bound on the mutual information rate between the input sequence x and the quantized output sequence r of the system in Figure 14, based on our MSE

Given R*xx* under a power constraint tr(R*xx*) ≤ *P*Tr, we choose x to be Gaussian, which is not necessarily the capacity achieving distribution for our quantized system. Then, we can obtain

*I*(x, r) = log2 |R*xx*| − *h*(x|r)

≥ log2

*<sup>I</sup>*(x, <sup>r</sup>) (<sup>1</sup> <sup>−</sup> *<sup>ρ</sup>q*)tr[R*xx*HHR−<sup>1</sup>

= log2 |R*xx*| − *h*(x − xˆ|r)

**<sup>I</sup>**−(1−*ρq*)R*xy*(R*yy*−*ρq*nondiag(R*yy*))−<sup>1</sup>


Since conditioning reduces entropy, we obtain inequality (31). On the other hand, the second term in (31) is upper bounded by the entropy of a Gaussian random variable whose covariance is equal to the error covariance matrix R*εε* of the linear MMSE estimate of x. Finally, we get

Considering the case of low SNR values, we get easily with R*yy* ≈ R*ηη*, (33) and (28), the

Compared with the mutual information *I*(x,y) for the unquantized case, also at low SNR [39], the mutual information for the quantized channel degrades only by the factor (1 − *ρq*). For

approach. Generally, the mutual information of this channel can be expressed as [26]

GWFQdiag(R*yy*)G<sup>H</sup>

WFQ 

*I*(x, r) = *H*(x) − *H*(x|r). (30)

≥ log2 |R*xx*| − *h*(x − xˆ) (31)

. (32)

H 

*ηη* H]/ log(2). (34)

*<sup>π</sup>* (see [35]) and the degradation of the mutual

. (33)

> 0, (29)

Let us examine the first derivative of the MSEWFQ in (26) with respect to *ρ<sup>q</sup>*

= tr 

*∂*MSEWFQ *∂ρq*

designs are jointly optimum with respect to the total MSE.

*3.2.4. Lower bound on the mutual information and the capacity*

a lower bound for *I*(x, r) (in bit/transmission) as

*I*(x, r) − log2

the spacial case *<sup>b</sup>* <sup>=</sup> 1, we have *<sup>ρ</sup>q*|*b*=<sup>1</sup> <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>2</sup>

<sup>2</sup> We assume also that *<sup>ρ</sup><sup>q</sup>* � 1 (or <sup>R</sup>*ηη* is diagonal). <sup>3</sup> Note that log <sup>|</sup>**<sup>I</sup>** <sup>+</sup> <sup>Δ</sup>X| ≈ tr(ΔX).

 

following first order approximation of the mutual information2 3

using (26) and (28)

$$\lim\_{\mathbf{x}\in\mathbb{N}\to 0} \frac{I(x,r)}{I(x,y)}\Big|\_{b=1} \approx \frac{2}{\pi}.\tag{35}$$

In other words, the power penalty due to the 1-bit quantization is approximately equal *<sup>π</sup>* <sup>2</sup> (1.96 dB) at low SNR. This shows that mono-bit ADCs may be used to save system power without an excessive degradation in performance, and confirms the significant potential of the coarsely quantized UWB MIMO channel. Using a different approach, [40] presented a similar result, and showed that the above approximation is asymptotically exact.

### *3.2.5. Simulation results*

The performance of the modified Wiener filter for a 4- and 5-bit quantized output MIMO system (WFQ), in terms of BER averaged over 1000 channel realizations, is shown in Figure 15 for a 10×10 MIMO system (QPSK), compared with the conventional Wiener filter (WF) and Zero-forcing filter (ZF). The symbols and the noise samples are assumed to be uncorrelated, that is, R*xx* = *σ*<sup>2</sup> *<sup>x</sup>* **I** and R*ηη* = *σ*<sup>2</sup> *<sup>η</sup>* **I**. Hereby, the SNR (in dB) is defined as

$$\text{SNR} = 10 \cdot \log\_{10} \left( \frac{\sigma\_x^2}{\sigma\_\eta^2} \right) . \tag{36}$$

Furthermore, we used a generic channel model, where the entries of H are complex-valued realizations of independent zero-mean Gaussian random variables with unit variance. Clearly, the modified Wiener filter outperforms the conventional Wiener filter at high SNR. This is because the effect of the quantization error is more pronounced at higher SNR values when compared to the additive Gaussian noise variance. Since the conventional Wiener filter converges to the ZF-filter at high SNR values and loses its regularized structure, its performance degrades asymptotically to the performance of the ZF-filter, when operating on quantized data. For comparison, we also plot the BER curves for the WF and ZF filter, for the case when no quantization is applied.

### **3.3. Channel estimation**

Because in general, the MIMO channel cannot be assumed known a-priori, a channel estimation has to be performed. In practice, it is highly desirable that the channel is estimated directly by the communication device – in our case by on-chip digital circuitry. However, this implies that the channel estimator is restricted to use received signal samples of a pilot sequence after single-bit quantization in the extreme case. This motivates investigation of channel estimation with coarse quantization. This problem was first addressed by [27], where a maximum likelihood (ML) channel estimation with quantized observation is presented. In general, the solution cannot be given in closed form, but requires an iterative numerical approach, which hampers the analysis of performance.

In [28], it has been shown that – in contrast to unquantized channel estimation – different orthogonal pilot sequences (with same average total transmit power and same length) yield different performances. Especially, orthogonality in the time-domain (time-multiplexed pilots) can be preferable to orthogonality in space. With orthogonal pilots

**4. Efficient digital hardware architecture**

check (bit) node.

*m* rows

¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨

*H*

*dV dC*

§

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*n* columns

 **0 0 1 0 1 1 0 1 1 0 0 1 0 1 1 0 1 0 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 0 1 0 1 0 1 0**

*m* check nodes

*n* bit nodes

Sole optimization of transmitting power in the standardization and conception phase of communication channels results in highly complex and energy-intensive receivers with a complex channel decoder as one of its key components. Neglecting the energy dissipation of the integrated decoder in this early phase results in suboptimal and, thus, costly communication systems in terms of manufacturing and usage costs. In the previous part of this chapter approaches to reduce the ADC complexity and, thus, the complexity of the subsequent digital components by using single-bit or medium-low resolution quantizations have been discussed. A quantitative comparison of these new approaches to standard receivers requires accurate cost models of the digital components. Quite accurate cost models are available for most of the communication system components except for channel decoders. While such cost models can be easily derived for Viterbi, Reed-Solomon, and Turbo Decoders, an estimation of the silicon area and the energy dissipation of LDPC decoders is challenging

Although LDPC codes have already been introduced by Gallager in 1962 [55], up to now they are known to achieve the best decoding performance [56] and are adopted in various communication standards (e.g. [57],[58],[59]) and other applications such as hard-disk drives [60]. They belong to the class of block codes and, thus, can be defined by a parity-check matrix *H* with *m* rows and *n* columns or by the corresponding Tanner Graph. Both are shown in Figure 16 for a very simplified LDPC code. Each row of the parity-check matrix represents one parity check wherein a '1'-entry in column *i* and row *j* indicates, that the received symbol *i* takes part in parity check number *j*. In the Tanner Graph such a parity check is represented by one so called check node and each column by one bit node. Furthermore, the number of one entries per row *dC* (column *dV*) defines the number of connected bit (check) nodes per

*L(Q L(ci) i)*

*L(ri,j)*

other check node

*-*

*-*

*L(Ri)*


*2·atanh ( ex )*

Chip-to-Chip and On-Chip Communications 95

*2·atanh ( ex )*

*2·atanh ( ex )*

other bit node




*L(ci)*

*log ( tanh ( x/2 ) )*

*log ( tanh ( x/2 ) )*

*log ( tanh ( x/2 ) )*

*L(qi,j)*

other check node

other bit node

¸ ¸ ¸ ¸ ¸ ¸ ¸ ¸

·

¹

**Figure 16.** Parity-check matrix, Tanner-Graph and non-linear recursive decoder loop

due to the high internal communication effort between the basic components.

**Figure 15.** The WFQ vs. the conventional WF and ZF receivers, QPSK modulation with *M* = 10, *N* = 10, 4- (*ρ<sup>q</sup>* = 0.01154) and 5- (*ρ<sup>q</sup>* = 0.00349) bit uniform quantizer.

that are multiplexed in time, the problem can be reduced from the MIMO to the SIMO (single-input-multiple-output) case, because each line of the multiconductor interconnect is excited separately for time-multiplexed pilots. Finally, the problem can be reduced to the SISO (single-input-single-output) case, when the channel estimation is performed separately in parallel at each receiving end of the multiconductor interconnect. For this case, in [28], a closed-form solution can be found for the maximum likelihood channel estimation problem, which makes performance analysis possible in an analytical fashion.

In [50], a more general setting for parameter estimation based on quantized observations was studied, which covers many processing tasks, e.g. channel estimation, synchronization, delay estimation, Direction Of Arrival (DOA) estimation, etc. An Expectation Maximization (EM) based algorithm is proposed to solve the Maximum a Posteriori Probability (MAP) estimation problem. Besides, the Cramér-Rao Bound (CRB) has been derived to analyze the estimation performance and its behavior with respect to the signal-to-noise ratio (SNR). The presented results treat both cases: pilot aided and non-pilot aided estimation. The paper extensively dealt with the extreme case of single bit quantization (comparator) which simplifies the sampling hardware considerably. It also focused on MIMO channel estimation and delay estimation as application area of the presented approach. Among others, a 2×2 channel estimation using 1-bit ADC is considered, which shows that reliable estimation may still be possible even when the quantization is very coarse, with any desired accuracy, provided the pilot sequence is long enough. Since in on-chip and chip-to-chip communications, the channel almost does not change in time, it is possible to use very long pilot sequences, and run the channel estimation only once, or once in a while.
