**2.1 Adaptive complex filtering**

As pointed out previously, adaptive complex filtering is a basic and very commonlyapplied DSP technique. An adaptive complex system consists of two basic building blocks: 12 Applications of Digital Signal Processing

*A real* DFT spectrum can be represented in a complex form. Forward *real* DFT results in cosine and sine wave terms, which then form respectively the real and imaginary parts of a complex number sequence. This substitution has the advantage of using powerful complex number math, but this is not true *complex* DFT. Despite the spectrum being in a complex form, the DFT remains *real* and *j* is not an integral part of the *complex representation* of *real*

Another mathematical inconvenience of *real* DFT is the absence of symmetry between analysis and synthesis equations, which is due to the exclusion of negative frequencies. In order to achieve a perfect reconstruction of the time domain signal, the first and last samples of the *real* DFT frequency spectrum, relating to zero frequency and Nyquist's frequency respectively, must have a scaling factor of 1/*N* applied to them rather than the 2/*N* used for

In contrast, *complex* DFT doesn't require a scaling factor of 2 as each value in the time domain corresponds to two spectral values located in a positive and a negative frequency; each one contributing half the time domain waveform amplitude, as shown in Fig. 7. The factor of 1/*N* is applied equally to all samples in the frequency domain. Taking the negative frequencies into account, *complex* DFT achieves a mathematically-favoured symmetry

*Complex* DFT overcomes the theoretical imperfections of *real* DFT in a manner helpful to other basic DSP transforms, such as forward and inverse z-transforms. A bright future is confidently predicted for *complex* DSP in general and the *complex* versions of Fourier

DSP is making a significant contribution to progress in many diverse areas of human endeavour – science, industry, communications, health care, security and safety, commercial

Based on powerful scientific mathematical principles, *complex* DSP has overlapping boundaries with the theory of, and is needed for many applications in, telecommunications.

Modern telecommunications very often uses narrowband signals, such as NBI (Narrowband Interference), RFI (Radio Frequency Interference), etc. These signals are complex by nature and hence it is natural for *complex* DSP techniques to be used to process them (Ovtcharov et

Telecommunication systems very commonly require processing to occur in real time, adaptive complex filtering being amongst the most frequently-used *complex* DSP techniques. When multiple communication channels are to be manipulated simultaneously, parallel

An efficient Adaptive Complex Filter Bank (ACFB) scheme is presented here, together with a short exploration of its application for the mitigation of narrowband interference signals in

As pointed out previously, adaptive complex filtering is a basic and very commonlyapplied DSP technique. An adaptive complex system consists of two basic building blocks:

between *forward* and *inverse* equations, i.e. between time and frequency domains.

**2. Complex DSP – some applications in telecommunications** 

This chapter presents a short exploration of precisely this common area.

processing systems are indicated (Nikolova et al, 2006), (Iliev et al, 2009).

MIMO (Multiple-Input Multiple-Output) communication systems.

DFT.

the rest of the samples.

transforms in particular.

business, space technologies etc.

al, 2009), (Nikolova et al, 2010).

**2.1 Adaptive complex filtering** 

the variable complex filter and the adaptive algorithm. Fig. 8 shows such a system based on a variable complex filter section designated LS1 (Low Sensitivity). The variable complex LS1 filter changes the central frequency and bandwidth independently (Iliev et al, 2002), (Iliev et al, 2006). The central frequency can be tuned by trimming the coefficient T, whereas the single coefficient E adjusts the bandwidth. The LS1 variable complex filter has two very important advantages: firstly, an extremely low passband sensitivity, which offers resistance to quantization effects and secondly, independent control of both central frequency and bandwidth over a wide frequency range.

The adaptive complex system (Fig.8) has a complex input *x*(*n*)=*xR*(*n*)+*jxI*(*n*) and provides both band-pass (BP) and band-stop (BS) complex filtering. The real and imaginary parts of the BP filter are respectively *yR*(*n*) and *yI*(*n*), whilst those of the BS filter are *eR*(*n*) and *eI*(*n*). The cost-function is the power of the BP/BS filter's output signal.

The filter coefficient T, responsible for the central frequency, is updated by applying an adaptive algorithm, for example LMS (Least Mean Square):

$$
\theta(n+1) = \theta(n) + \mu \operatorname{Re} [e(n)y'^\*(n)].\tag{11}
$$

The step sizeP controls the speed of convergence, ( ) denotes complex-conjugate, *y*<sup>c</sup> (*n*) is the derivative of complex BP filter output *y*(*n*) with respect to the coefficient, which is subject to adaptation.

Fig. 8. Block-diagram of an LS1-based adaptive complex system

In order to ensure the stability of the adaptive algorithm, the range of the step size P should be set according to (Douglas, 1999):

$$0 < \mu < \frac{P}{N\sigma^2}.\tag{11}$$

where *N* is the filter order, ǔ2 is the power of the signal *y*<sup>c</sup> (*n*) and *P* is a constant which depends on the statistical characteristics of the input signal. In most practical situations, *P* is approximately equal to 0.1.

Complex Digital Signal Processing in Telecommunications 15

The experiments are carried out with an input signal composed of three complex sinesignals of different frequencies, mixed with white noise. Fig. 10 displays learning curves for the coefficientsT1, T2 and T3. The ACFB shows the high efficacy of the parallel filtering process. The main advantages of both the adaptive filter structure and the ACFB lie in their

low computational complexity and rapid convergence of adaptation.

Fig. 10. Learning curves of an ACFB consisting of three complex LS1-sections

**filtering** 

complex filtering in the frequency domain.

sequence spreading approaches.

**2.2.1 NBI Suppression in UWB MIMO systems** 

**2.2 Narrowband interference suppression for MIMO systems using adaptive complex** 

The sub-sections which follow examine the problem of narrowband interference in two particular MIMO telecommunication systems. Different NBI suppression methods are observed and experimentally compared to the *complex* DSP technique using adaptive

Ultrawideband (UWB) systems show excellent potential benefits when used in the design of high-speed digital wireless home networks. Depending on how the available bandwidth of the system is used, UWB can be divided into two groups: *single-band* and *multi-band* (MB). Conventional UWB technology is based on *single-band* systems and employs carrier-free communications. It is implemented by directly modulating information into a sequence of impulse-like waveforms; support for multiple users is by means of time-hopping or direct

The UWB frequency band of *multi-band* UWB systems is divided into several sub-bands. By interleaving the symbols across sub-bands, multi-band UWB can maintain the power of the transmission as though a wide bandwidth were being utilized. The advantage of the multiband approach is that it allows information to be processed over a much smaller bandwidth, thereby reducing overall design complexity as well as improving spectral flexibility and worldwide adherence to the relevant standards. The constantly-increasing demand for higher data transmission rates can be satisfied by exploiting both multipath- and spatialdiversity, using MIMO together with the appropriate modulation and coding techniques

The very low sensitivity of the variable complex LS1 filter section ensures the general efficiency of the adaptation and a high tuning accuracy, even with severely quantized multiplier coefficients.

This approach can easily be extended to the adaptive complex filter bank synthesis in parallel complex signal processing.

In (Nikolova et al, 2002) a narrowband ACFB is designed for the detection of multiple complex sinusoids. The filter bank, composed of three variable complex filter sections, is aimed at detecting multiple complex signals (Fig. 9).

Fig. 9. Block-diagram of an adaptive complex filter bank system

14 Applications of Digital Signal Processing

The very low sensitivity of the variable complex LS1 filter section ensures the general efficiency of the adaptation and a high tuning accuracy, even with severely quantized

This approach can easily be extended to the adaptive complex filter bank synthesis in

In (Nikolova et al, 2002) a narrowband ACFB is designed for the detection of multiple complex sinusoids. The filter bank, composed of three variable complex filter sections, is

E

E

E

E

E

E

*yI*1(*n*)

*yI*2(*n*)

*yI*3(*n*)

*yR*3(*n*)

*yR*2(*n*)

*yR*1(*n*)

*eR*(*n*)

*eI*(*n*)

multiplier coefficients.

*xI*(*n*)

*xR*(*n*)

parallel complex signal processing.

aimed at detecting multiple complex signals (Fig. 9).

sinT

cosT

sinT

cosT

sinT

cosT

cosT

cosT

cosT

z-1

sinT

z-1

z-1

sinT

z-1

z-1

sinT

z-1

Adaptive Algoritm

Fig. 9. Block-diagram of an adaptive complex filter bank system

The experiments are carried out with an input signal composed of three complex sinesignals of different frequencies, mixed with white noise. Fig. 10 displays learning curves for the coefficientsT1, T2 and T3. The ACFB shows the high efficacy of the parallel filtering process. The main advantages of both the adaptive filter structure and the ACFB lie in their low computational complexity and rapid convergence of adaptation.

Fig. 10. Learning curves of an ACFB consisting of three complex LS1-sections
