**1.1 Complex DSP versus real DSP**

Digital Signal Processing (DSP) is a vital tool for scientists and engineers, as it is of fundamental importance in many areas of engineering practice and scientific research.

The "alphabet" of DSP is mathematics and although most practical DSP problems can be solved by using real number mathematics, there are many others which can only be satisfactorily resolved or adequately described by means of complex numbers.

If real number mathematics is the language of *real* DSP, then complex number mathematics is the language of *complex* DSP. In the same way that real numbers are a part of complex numbers in mathematics, *real* DSP can be regarded as a part of *complex* DSP (Smith, 1999).

Complex mathematics manipulates complex numbers – the representation of two variables as a single number - and it may appear that *complex* DSP has no obvious connection with our everyday experience, especially since many DSP problems are explained mainly by means of real number mathematics. Nonetheless, some DSP techniques are based on complex mathematics, such as Fast Fourier Transform (FFT), z-transform, representation of periodical signals and linear systems, etc. However, the imaginary part of complex transformations is usually ignored or regarded as zero due to the inability to provide a readily comprehensible physical explanation.

One well-known practical approach to the representation of an engineering problem by means of complex numbers can be referred to as the *assembling approach*: the real and imaginary parts of a complex number are real variables and individually can represent two real physical parameters. Complex math techniques are used to process this complex entity once it is assembled. The real and imaginary parts of the resulting complex variable preserve the same real physical parameters. This approach is not universally-applicable and can only be used with problems and applications which conform to the requirements of complex math techniques. Making a complex number entirely mathematically equivalent to a substantial physical problem is the real essence of *complex* DSP. Like complex Fourier transforms, complex DSP transforms show the fundamental nature of *complex* DSP and such complex techniques often increase the power of basic DSP methods. The development and application of *complex* DSP are only just beginning to increase and for this reason some researchers have named it *theoretical* DSP.

Complex Digital Signal Processing in Telecommunications 5

    0 0

*R I x n Ae n x n Ae n*

Clearly, *xR*(*n*) and *xI*(*n*) are real discrete-time sinusoidal signals whose amplitude \_A\_*e*

(a)

(b)

(c)

Fig. 1. Complex exponential signal *x*(*n*) and its real and imaginary components *xR*(*n*) and

Sample number

Sample number

Sample number

I

V

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> -0.5

Sample number

Real part Imaginary part

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> -60

Sample number

Real part Imaginary part

0 10 20 30 40 50 60

Sample number

*xI*(*n*) for (a) V0=-0.085; (b) V0=0.085 and (c) V0=0

Real part Imaginary part



Amplitude

Amplitude

Amplitude

Z

constant (V0=0), increasing (V0>0) or decreasing V0<0) exponents (Fig. 1).

0 0 cos ; sin . *n n*

 V




 Z


Real part Imaginary part

Imaginary part Real part


Real part Imaginary part

0


0

0.5

I

(3)

<sup>V</sup>o*<sup>n</sup>* is

0.5

It is evident that *complex* DSP is more complicated than *real* DSP. Complex DSP transforms are highly theoretical and mathematical; to use them efficiently and professionally requires a large amount of mathematics study and practical experience.

Complex math makes the mathematical expressions used in DSP more compact and solves the problems which real math cannot deal with. Complex DSP techniques can complement our understanding of how physical systems perform but to achieve this, we are faced with the necessity of dealing with extensive sophisticated mathematics. For DSP professionals there comes a point at which they have no real choice since the study of complex number mathematics is the foundation of DSP.

#### **1.2 Complex representation of signals and systems**

All naturally-occurring signals are real; however in some signal processing applications it is convenient to represent a signal as a complex-valued function of an independent variable. For purely mathematical reasons, the concept of complex number representation is closely connected with many of the basics of electrical engineering theory, such as voltage, current, impedance, frequency response, transfer-function, Fourier and z-transforms, etc.

*Complex* DSP has many areas of application, one of the most important being modern telecommunications, which very often uses narrowband analytical signals; these are complex in nature (Martin, 2003). In this field, the complex representation of signals is very useful as it provides a simple interpretation and realization of complicated processing tasks, such as modulation, sampling or quantization.

It should be remembered that a complex number could be expressed in *rectangular*, *polar* and *exponential* forms:

  cos sin *<sup>j</sup> a jb A j Ae* T T T. (1)

The third notation of the complex number in the equation (1) is referred to as *complex exponential* and is obtained after Euler's relation is applied. The exponential form of complex numbers is at the core of *complex* DSP and enables magnitude *A* and phase lj components to be easily derived.

Complex numbers offer a compact representation of the most often-used waveforms in signal processing – *sine* and *cosine* waves (Proakis & Manolakis, 2006). The complex number representation of sinusoids is an elegant technique in signal and circuit analysis and synthesis, applicable when the rules of complex math techniques coincide with those of sine and cosine functions. Sinusoids are represented by complex numbers; these are then processed mathematically and the resulting complex numbers correspond to sinusoids, which match the way sine and cosine waves would perform if they were manipulated individually. The complex representation technique is possible only for sine and cosine waves of the same frequency, manipulated mathematically by linear systems.

The use of Euler's identity results in the class of complex exponential signals:

$$\mathbf{x}\left(n\right) = A\alpha^n = \left| A \right| e^{j\phi} e^{\left(\sigma\_0 + j\alpha\_0\right)} = \mathbf{x}\_R\left(n\right) + j\mathbf{x}\_I\left(n\right) \,. \tag{2}$$

  0 0 *<sup>j</sup> e* V Z D and *<sup>j</sup> A Ae* Iare complex numbers thus obtaining: 4 Applications of Digital Signal Processing

It is evident that *complex* DSP is more complicated than *real* DSP. Complex DSP transforms are highly theoretical and mathematical; to use them efficiently and professionally requires

Complex math makes the mathematical expressions used in DSP more compact and solves the problems which real math cannot deal with. Complex DSP techniques can complement our understanding of how physical systems perform but to achieve this, we are faced with the necessity of dealing with extensive sophisticated mathematics. For DSP professionals there comes a point at which they have no real choice since the study of complex number

All naturally-occurring signals are real; however in some signal processing applications it is convenient to represent a signal as a complex-valued function of an independent variable. For purely mathematical reasons, the concept of complex number representation is closely connected with many of the basics of electrical engineering theory, such as voltage, current,

*Complex* DSP has many areas of application, one of the most important being modern telecommunications, which very often uses narrowband analytical signals; these are complex in nature (Martin, 2003). In this field, the complex representation of signals is very useful as it provides a simple interpretation and realization of complicated processing tasks,

It should be remembered that a complex number could be expressed in *rectangular*, *polar* and

 cos sin *<sup>j</sup> a jb A j Ae*

 T

The third notation of the complex number in the equation (1) is referred to as *complex exponential* and is obtained after Euler's relation is applied. The exponential form of complex numbers is at the core of *complex* DSP and enables magnitude *A* and phase lj components to

Complex numbers offer a compact representation of the most often-used waveforms in signal processing – *sine* and *cosine* waves (Proakis & Manolakis, 2006). The complex number representation of sinusoids is an elegant technique in signal and circuit analysis and synthesis, applicable when the rules of complex math techniques coincide with those of sine and cosine functions. Sinusoids are represented by complex numbers; these are then processed mathematically and the resulting complex numbers correspond to sinusoids, which match the way sine and cosine waves would perform if they were manipulated individually. The complex representation technique is possible only for sine and cosine

> 0 0 *<sup>n</sup> <sup>j</sup> <sup>j</sup> R I x n A Ae e x n jx n* I V Z

are complex numbers thus obtaining:

waves of the same frequency, manipulated mathematically by linear systems. The use of Euler's identity results in the class of complex exponential signals:

D

I

 T T

. (2)

. (1)

impedance, frequency response, transfer-function, Fourier and z-transforms, etc.

a large amount of mathematics study and practical experience.

**1.2 Complex representation of signals and systems** 

such as modulation, sampling or quantization.

*exponential* forms:

be easily derived.

  0 0 *<sup>j</sup> e* V Z

and *<sup>j</sup> A Ae*

D

mathematics is the foundation of DSP.

$$\mathbf{x}\_R(n) = \left| A \right| e^{\sigma\_0 n} \cos \left( a \rho\_0 n + \phi \right); \qquad \mathbf{x}\_I(n) = \left| A \right| e^{\sigma\_0 n} \sin \left( a \rho\_0 n + \phi \right). \tag{3}$$

Clearly, *xR*(*n*) and *xI*(*n*) are real discrete-time sinusoidal signals whose amplitude \_A\_*e* <sup>V</sup>o*<sup>n</sup>* is constant (V0=0), increasing (V0>0) or decreasing V0<0) exponents (Fig. 1).

Fig. 1. Complex exponential signal *x*(*n*) and its real and imaginary components *xR*(*n*) and *xI*(*n*) for (a) V0=-0.085; (b) V0=0.085 and (c) V0=0

Complex Digital Signal Processing in Telecommunications 7

 *<sup>j</sup> <sup>n</sup> <sup>j</sup> <sup>n</sup> <sup>j</sup> <sup>n</sup> X e X e jX e CR I* ZZ

The real signal and its Hilbert transform are respectively the real and imaginary parts of the

*XR*(*ej* Z*n*)

*jXI*(*ej* Z*n*)



*XC*(*ej* Z*n*)




shifted by S/2 for positive frequencies and by –S/2 for negative frequencies, thus the

phase shifted for negative frequencies – the solid blue line (Fig. 3b). The complex signal

multiplied by *j* (square root of -1), are identical for positive frequencies and –S/2

occupies half of the real signal frequency band; its amplitude is the sum of the


Fig. 3. Complex signal derivation using the Hilbert transformation

pattern areas in Fig. 3b are obtained. The real signal  *<sup>j</sup> <sup>n</sup> X e <sup>R</sup>*

 *<sup>j</sup> <sup>n</sup> X e <sup>I</sup>* Z

 *<sup>j</sup> <sup>n</sup> X e <sup>C</sup>* Z

 *<sup>j</sup> <sup>n</sup> X e <sup>R</sup>* Z

 and  *<sup>j</sup> <sup>n</sup> <sup>I</sup> jX e* Z

Z

analytic signal *<sup>C</sup> <sup>j</sup> <sup>n</sup> X e*

According to the Hilbert transformation, the components of the  *<sup>j</sup> <sup>n</sup> X e <sup>R</sup>*

is depicted in Fig. 3d.

*XC\**(*e-j*<sup>Z</sup>*n*)

analytic signal; these have the same amplitude and S/2 phase-shift (Fig. 3).

 Z

. (7)

Z

(a)

(b)

(c)

(d)

Z

and the imaginary one

spectrum are

Z

Z

Z

Z

amplitudes (Fig. 3c). The spectrum of the complex conjugate

The spectrum of a real discrete-time signal lies between –ǚ*s*/2 and ǚ*s*/2 (ǚ*s* is the sampling frequency in radians per sample), while the spectrum of a complex signal is twice as narrow and is located within the positive frequency range only.

Narrowband signals are of great use in telecommunications. The determination of a signal's attributes, such as frequency, envelope, amplitude and phase are of great importance for signal processing e.g. modulation, multiplexing, signal detection, frequency transformation, etc. These attributes are easier to quantify for narrowband signals than for wideband signals (Fig. 2). This makes narrowband signals much simpler to represent as complex signals.

Fig. 2. Narrowband signal (a) *xn n* <sup>1</sup>  sin 60 4 cos 2 S S S *n* ; wideband signal (b) *xn n* <sup>2</sup>  sin 60 4 cos 16 S S S *n* 

Over the years different techniques of describing narrowband complex signals have been developed. These techniques differ from each other in the way the imaginary component is derived; the real component of the complex representation is the real signal itself.

Some authors (Fink, 1984) suggest that the imaginary part of a complex narrowband signal can be obtained from the first *x n* c *<sup>R</sup>*  and second *x n* cc*<sup>R</sup>*  derivatives of the real signal:

$$\mathbf{x}\_{R}(n) = -\mathbf{x}\_{R}'(n)\sqrt{\frac{-\mathbf{x}\_{R}(n)}{\mathbf{x}\_{R}''(n)}}\,\mathrm{}.\tag{4}$$

One disadvantage of the representation in equation (4) is that insignificant changes in the real signal *xR*(*n*) can alter the imaginary part *xI*(*n*) significantly; furthermore the second derivative can change its sign, thus removing the sense of the square root.

Another approach to deriving the imaginary component of a complex signal representation, applicable to harmonic signals, is as follows (Gallagher, 1968):

$$
\lambda \mathbf{x}\_I(n) = \frac{-\mathbf{x}\_R(n)}{a\_0},
\tag{5}
$$

where Z0 is the frequency of the real harmonic signal.

Analytical representation is another well-known approach used to obtain the imaginary part of a complex signal, named the *analytic* signal. An analytic complex signal is represented by its *inphase* (the real component) and *quadrature* (the imaginary component). The approach includes a low-frequency envelope modulation using a complex carrier signal – a complex exponent 0 *<sup>j</sup> <sup>n</sup> e* Znamed *cissoid* (Crystal & Ehrman, 1968) or *complexoid* (Martin, 2003):

$$\mathbf{x}\_{\mathbb{R}}(n) \otimes e^{i\mathbf{a}\_{0}\mathbf{u}} \implies \mathbf{x}(n) = \mathbf{x}\_{\mathbb{R}}(n) e^{i\mathbf{a}\_{0}\mathbf{u}} = \mathbf{x}\_{\mathbb{R}}(n) \left[ \cos a\_{0}\mathbf{u} + j \sin a\_{0}\mathbf{u} \right] = \mathbf{x}\_{\mathbb{R}}(n) + j \mathbf{x}\_{\mathbb{I}}(n) \,. \tag{6}$$

In the frequency domain an analytic complex signal is:

6 Applications of Digital Signal Processing

The spectrum of a real discrete-time signal lies between –ǚ*s*/2 and ǚ*s*/2 (ǚ*s* is the sampling frequency in radians per sample), while the spectrum of a complex signal is twice as narrow

Narrowband signals are of great use in telecommunications. The determination of a signal's attributes, such as frequency, envelope, amplitude and phase are of great importance for signal processing e.g. modulation, multiplexing, signal detection, frequency transformation, etc. These attributes are easier to quantify for narrowband signals than for wideband signals (Fig. 2). This makes narrowband signals much simpler to represent as complex signals.

> -0.5 0 0.5 1

> > S

S

 S *n* ;

 S *n* 

> *R*

*R x n*

*x n*

Amplitude

(a) (b)

Over the years different techniques of describing narrowband complex signals have been developed. These techniques differ from each other in the way the imaginary component is

Some authors (Fink, 1984) suggest that the imaginary part of a complex narrowband signal

 

One disadvantage of the representation in equation (4) is that insignificant changes in the real signal *xR*(*n*) can alter the imaginary part *xI*(*n*) significantly; furthermore the second

Another approach to deriving the imaginary component of a complex signal representation,

 

Analytical representation is another well-known approach used to obtain the imaginary part of a complex signal, named the *analytic* signal. An analytic complex signal is represented by its *inphase* (the real component) and *quadrature* (the imaginary component). The approach includes a low-frequency envelope modulation using a complex carrier signal – a complex

0 *R*

Z

named *cissoid* (Crystal & Ehrman, 1968) or *complexoid* (Martin, 2003):

Z

 Z

. (6)

   > @   0 0

0 0 cos sin *j n j n R R R R <sup>I</sup> x n e x n x n e x n n j n x n jx n*

 Z *x n*

S

S

derived; the real component of the complex representation is the real signal itself.

*I R*

*I*

*x n*

derivative can change its sign, thus removing the sense of the square root.

applicable to harmonic signals, is as follows (Gallagher, 1968):

where Z0 is the frequency of the real harmonic signal.

In the frequency domain an analytic complex signal is:

*xn x n*

*<sup>R</sup>*  and second *x n* cc

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> -1

Wideband signal x2(n)

Sample number

*<sup>R</sup>*  derivatives of the real signal:

, (5)

<sup>c</sup> cc . (4)

and is located within the positive frequency range only.

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> -1

Narrowband signal x1(n)

Sample number

can be obtained from the first *x n* c

Fig. 2. Narrowband signal (a) *xn n* <sup>1</sup>  sin 60 4 cos 2

wideband signal (b) *xn n* <sup>2</sup>  sin 60 4 cos 16


exponent 0 *<sup>j</sup> <sup>n</sup> e*

Z

Z

Amplitude

$$X\_C\left(e^{j\alpha m}\right) = X\_R\left(e^{j\alpha m}\right) + jX\_I\left(e^{j\alpha m}\right). \tag{7}$$

The real signal and its Hilbert transform are respectively the real and imaginary parts of the analytic signal; these have the same amplitude and S/2 phase-shift (Fig. 3).

Fig. 3. Complex signal derivation using the Hilbert transformation

According to the Hilbert transformation, the components of the  *<sup>j</sup> <sup>n</sup> X e <sup>R</sup>* Z spectrum are shifted by S/2 for positive frequencies and by –S/2 for negative frequencies, thus the pattern areas in Fig. 3b are obtained. The real signal  *<sup>j</sup> <sup>n</sup> X e <sup>R</sup>* Z and the imaginary one *<sup>j</sup> <sup>n</sup> X e <sup>I</sup>* Z multiplied by *j* (square root of -1), are identical for positive frequencies and –S/2 phase shifted for negative frequencies – the solid blue line (Fig. 3b). The complex signal *<sup>j</sup> <sup>n</sup> X e <sup>C</sup>* Z occupies half of the real signal frequency band; its amplitude is the sum of the *<sup>j</sup> <sup>n</sup> X e <sup>R</sup>* Z and  *<sup>j</sup> <sup>n</sup> <sup>I</sup> jX e* Z amplitudes (Fig. 3c). The spectrum of the complex conjugate analytic signal *<sup>C</sup> <sup>j</sup> <sup>n</sup> X e* Zis depicted in Fig. 3d.

Complex Digital Signal Processing in Telecommunications 9

Digital systems and signals can be represented in three domains – time domain, z-domain and frequency domain. To cross from one domain to another, the Fourier and z-transforms are employed (Fig. 5). Both transforms are fundamental building-blocks of signal processing

> **Time Domain**

The Fourier transforms group contains four families, which differ from one another in the type of time-domain signal which they process - *periodic* or *aperiodic* and *discrete* or *continuous*. Discrete Fourier Transform (DFT) deals with *discrete periodic* signals, Discrete Time Fourier Transform (DTFT) with *discrete aperiodic* signals, and Fourier Series and Fourier Transform with *periodic* and *aperiodic continuous* signals respectively. In addition to having forward and inverse versions, each of these four Fourier families exists in two forms - *real* and *complex*, depending on whether real or complex number math is used. All four Fourier transform families decompose signals into sine and cosine waves; when these are expressed by complex number equations, using Euler's identity, the *complex* versions of

DFT is the most often-used Fourier transform in DSP. The DFT family is a basic mathematical tool in various processing techniques performed in the frequency domain, for instance frequency analysis of digital systems and spectral representation of discrete signals. In this chapter, the focus is on *complex* DFT. This is more sophisticated and wide-ranging than real DFT, but is based on the more complicated complex number math. However, numerous digital signal processing techniques, such as convolution, modulation, compression, aliasing, etc. can be better described and appreciated via this extended math.

*Complex* DFT equations are shown in Table 1. The *forward complex* DFT equation is also called *analysis* equation. This calculates the frequency domain values of the discrete periodic signal, whereas the *inverse* (*synthesis*) equation computes the values in the time domain.

The time domain signal *x*(*n*) is a complex discrete periodic signal; only an *N*-point unique discrete sequence from this signal, situated in a single time-interval (0÷*N*, -*N*/2÷*N*/2, etc.) is

Ztransforms

**Z-Domain**

**1.3 Complex digital processing techniques - complex Fourier transforms** 

theory and exist in two formats - *forward* and *inverse* (Smith, 1999).

Fourier transforms

Fig. 5. Relationships between frequency, time, and z- domains

**Frequency Domain**

(Sklar, 2001)

the Fourier transforms are introduced.

Table 1. Complex DFT transforms in rectangular form

In the frequency domain the analytic complex signal, its complex conjugate signal, real and imaginary components are related as follows:

$$\begin{aligned} X\_R\left(e^{jom}\right) &= \frac{1}{2} \left\{ X\left(e^{jom}\right) + X^\*\left(e^{-jom}\right) \right\} \\ jX\_I\left(e^{jom}\right) &= \frac{1}{2} \left\{ X\left(e^{jom}\right) - X^\*\left(e^{-jom}\right) \right\} \\ X\left(e^{jom}\right) &= \begin{cases} 2X\_R\left(e^{jom}\right) = 2jX\_I\left(e^{jom}\right), & 0 < o < o\_S/2 \\ 0, & -o\_S/2 \le o < 0 \end{cases} \end{aligned} \tag{8}$$

Discrete-time complex signals are easily processed by digital complex circuits, whose transfer functions contain complex coefficients (Márquez, 2011).

An output complex signal *YC* (*z*) is the response of a complex system with transfer function *HC* (*z*), when complex signal *XC* (*z*) is applied as an input. Being complex functions, *XC* (*z*), *YC* (*z*) and *HC* (*z*), can be represented by their real and imaginary parts:

$$\begin{array}{ccccc}\underbrace{Y\_{\mathbb{C}}(z)}\_{\mathbb{J}} &=& \underbrace{H\_{\mathbb{C}}(z)}\_{\mathbb{J}} & \underbrace{X\_{\mathbb{C}}(z)}\_{\mathbb{J}}\\\left[Y\_{\mathbb{R}}(z) + jY\_{I}(z)\right] &=& \left[H\_{\mathbb{R}}(z) + jH\_{I}(z)\right] & \left[X\_{\mathbb{R}}(z) + jX\_{I}(z)\right] \end{array} \tag{9}$$

After mathematical operations are applied, the complex output signal and its real and imaginary parts become:

$$\begin{aligned} \mathcal{Y}\_{\mathbb{C}}(z) &= \left[ H\_{\mathbb{R}}(z) + jH\_{I}(z) \right] \left[ X\_{\mathbb{R}}(z) + jX\_{I}(z) \right] = \\ &= \underbrace{\left[ H\_{\mathbb{R}}(z)X\_{\mathbb{R}}(z) - H\_{I}(z)X\_{I}(z) \right]}\_{\mathbb{\tilde{U}}} + \underbrace{j \left[ \underbrace{H\_{I}(z)X\_{\mathbb{R}}(z) + H\_{\mathbb{R}}(z)X\_{I}(z)}\_{\mathbb{\tilde{U}}} \right]}\_{\mathbb{\tilde{U}}} \end{aligned} + \underbrace{j \left[ \underbrace{H\_{I}(z)X\_{\mathbb{R}}(z) + H\_{\mathbb{R}}(z)X\_{I}(z)}\_{\mathbb{\tilde{U}}} \right]}\_{Y\_{I}(z)} \tag{10}$$

According to equation (10), the block-diagram of a complex system will be as shown in Fig. 4.

Fig. 4. Block-diagram of a complex system

8 Applications of Digital Signal Processing

In the frequency domain the analytic complex signal, its complex conjugate signal, real and

2 2 ,0 2

 Z

*S*

 

Z

Z Z

 ¼ ¬

(8)

(9)

¼ (10)

 Z

0, 2 0

° ® ° d ¯

*j n j n j n R I S*

Discrete-time complex signals are easily processed by digital complex circuits, whose

An output complex signal *YC* (*z*) is the response of a complex system with transfer function *HC* (*z*), when complex signal *XC* (*z*) is applied as an input. Being complex functions, *XC* (*z*),

 

 

*C CC*

*Y z H z X z*

*RI R I R I*

*Y z jY z H z jH z X z jX z* 

ª º ª ºª º ¬ ¼ ¬ ¼ ¬ ¼

After mathematical operations are applied, the complex output signal and its real and

 

 

*RR II IR RI*

ª º ª º ¬

*H zX z H zX z jH zX z H zX z*

*R I*

According to equation (10), the block-diagram of a complex system will be as shown in

*HR* (*z*)

*XI* (*z*) **+** *YI* (*z*)

*HI* (*z*)

*HI* (*z*)

*XR* (*z*) **+**

*Y z Y z* 

 

*HR* (*z*) *YR* (*z*)

ZZ

ZZ

 ^  `

*j n j n j n*

transfer functions contain complex coefficients (Márquez, 2011).

 

ª ºª º ¬ ¼ ¬ ¼

 

*C R IR I*

Fig. 4. Block-diagram of a complex system

*Y z H z jH z X z jX z*

*X e Xe X e*

*jX e X e X e*

 ^ `  

*j n j n j n*

 

Z

*X e jX e X e*

*YC* (*z*) and *HC* (*z*), can be represented by their real and imaginary parts:

imaginary components are related as follows:

Z

Z

Z

N

imaginary parts become:

Fig. 4.

*R*

*I*

#### **1.3 Complex digital processing techniques - complex Fourier transforms**

Digital systems and signals can be represented in three domains – time domain, z-domain and frequency domain. To cross from one domain to another, the Fourier and z-transforms are employed (Fig. 5). Both transforms are fundamental building-blocks of signal processing theory and exist in two formats - *forward* and *inverse* (Smith, 1999).

Fig. 5. Relationships between frequency, time, and z- domains

The Fourier transforms group contains four families, which differ from one another in the type of time-domain signal which they process - *periodic* or *aperiodic* and *discrete* or *continuous*. Discrete Fourier Transform (DFT) deals with *discrete periodic* signals, Discrete Time Fourier Transform (DTFT) with *discrete aperiodic* signals, and Fourier Series and Fourier Transform with *periodic* and *aperiodic continuous* signals respectively. In addition to having forward and inverse versions, each of these four Fourier families exists in two forms - *real* and *complex*, depending on whether real or complex number math is used. All four Fourier transform families decompose signals into sine and cosine waves; when these are expressed by complex number equations, using Euler's identity, the *complex* versions of the Fourier transforms are introduced.

DFT is the most often-used Fourier transform in DSP. The DFT family is a basic mathematical tool in various processing techniques performed in the frequency domain, for instance frequency analysis of digital systems and spectral representation of discrete signals. In this chapter, the focus is on *complex* DFT. This is more sophisticated and wide-ranging than real DFT, but is based on the more complicated complex number math. However, numerous digital signal processing techniques, such as convolution, modulation, compression, aliasing, etc. can be better described and appreciated via this extended math. (Sklar, 2001)

*Complex* DFT equations are shown in Table 1. The *forward complex* DFT equation is also called *analysis* equation. This calculates the frequency domain values of the discrete periodic signal, whereas the *inverse* (*synthesis*) equation computes the values in the time domain.


Table 1. Complex DFT transforms in rectangular form

The time domain signal *x*(*n*) is a complex discrete periodic signal; only an *N*-point unique discrete sequence from this signal, situated in a single time-interval (0÷*N*, -*N*/2÷*N*/2, etc.) is

Complex Digital Signal Processing in Telecommunications 11

frequency domain. The frequency range of its real, Re *X*(*k*), and imaginary part, Im *X*(*k*), comprises both positive and negative frequencies simultaneously. Since the considered time domain signal is real, Re *X*(*k*) is even (the spectral values *A* and *B* have the same sign), while

The amplitude of each of the four spectral peaks is *M*/2, which is half the amplitude of the time domain signal. The single frequency interval under consideration [-¼ǚs÷¼ǚs] ([-0.5÷0.5] when normalized frequency is used) is symmetric with respect to a frequency of zero. The real frequency spectrum Re *X*(*k*) is used to reconstruct a cosine time domain signal, whilst the imaginary spectrum Im *X*(*k*) results in a negative sine wave, both with amplitude *M* in accordance with the complex analysis equation (Table 1). In a way analogous to the example shown in Fig. 7, a complex frequency spectrum can also be

> Real time domain signal of frequency ǚ0

**Forward** *complex* **DFT**

**Complex spectrum** 

 

cos 2

*M*/2

0 0

*M*

*D*

 

0 0

cos 2

*M*

*n*

¿ ¾ ½

*n*

¿ ¾ ½

 *nMkX*

Z

*I* 0

sin)(

*n*

 ZZ 

*n*

ZZ


ǚ0 0

Imaginary part of complex spectrum Im X(k)

*<sup>M</sup> <sup>C</sup>*

¯ ®

sin 2

*C*


sin 2

*M*/2

*<sup>M</sup> <sup>D</sup>*

¯ ®

the imaginary part Im *X*(*k*) is odd (*C* is negative, *D* is positive).

 

0 0

*M*

*B*

*M*/2

 

0 0

sin 2

sin 2

*M*

*n*

Fig. 7. Inverse *complex* DFT - reconstruction of a real time domain signal Why is *complex* DFT used since it involves intricate complex number math?

frequencies are always encountered in conjunction with complex numbers.

¿ ¾ ½

> ¿ ¾ ½

*Complex* DFT has persuasive advantages over *real* DFT and is considered to be the more comprehensive version. *Real* DFT is mathematically simpler and offers practical solutions to real world problems; by extension, negative frequencies are disregarded. Negative

*n*

 *nMkX*

Z

*R* 0

cos)(

*n*

 ZZ 

*n*

Real part of complex spectrum Re X(k)

ZZ


*<sup>M</sup> <sup>A</sup>*

¯ ®

cos 2

cos 2

*A M*/2

*<sup>M</sup> <sup>B</sup>*

¯ ®

derived.

considered. The forward equation multiplies the periodic time domain number series from *x*(0) to *x*(*N*-1) by a sinusoid and sums the results over the complete time-period.

The frequency domain signal *X*(*k*) is an *N*-point complex periodic signal in a single frequency interval, such as [0÷0.5ǚs], [-0.5ǚs÷0], [-0.25ǚs÷0.25ǚs], etc. (the sampling frequency ǚs is often used in its normalized value). The inverse equation employs all the *N* points in the frequency domain to calculate a particular discrete value of the time domain signal. It is clear that *complex* DFT works with finite-length data.

Both the time domain *x*(*n*) and the frequency domain *X*(*k*) signals are complex numbers, i.e. *complex* DFT also recognizes negative time and negative frequencies. Complex mathematics accommodates these concepts, although imaginary time and frequency have only a theoretical existence so far. *Complex* DFT is a symmetrical and mathematically comprehensive processing technology because it doesn't discriminate between negative and positive frequencies.

Fig. 6 shows how the forward *complex* DFT algorithm works in the case of a complex timedomain signal. *xR*(*n*) is a real time domain signal whose frequency spectrum has an even real part and an odd imaginary part; conversely, the frequency spectrum of the imaginary part of the time domain signal *xI*(*n*) has an odd real part and an even imaginary part. However, as can be seen in Fig. 6, the actual frequency spectrum is the sum of the two individuallycalculated spectra. In reality, these two time domain signals are processed simultaneously, which is the whole point of the Fast Fourier Transform (FFT) algorithm.

Fig. 6. Forward *complex* DFT algorithm

The imaginary part of the time-domain complex signal can be omitted and the time domain then becomes totally real, as is assumed in the numerical example shown in Fig. 7. A real sinusoidal signal with amplitude *M,* represented in a complex form, contains a positive ǚ0 and a negative frequency -ǚ0. The complex spectrum *X*(*k*) describes the signal in the 10 Applications of Digital Signal Processing

considered. The forward equation multiplies the periodic time domain number series from

The frequency domain signal *X*(*k*) is an *N*-point complex periodic signal in a single frequency interval, such as [0÷0.5ǚs], [-0.5ǚs÷0], [-0.25ǚs÷0.25ǚs], etc. (the sampling frequency ǚs is often used in its normalized value). The inverse equation employs all the *N* points in the frequency domain to calculate a particular discrete value of the time domain

Both the time domain *x*(*n*) and the frequency domain *X*(*k*) signals are complex numbers, i.e. *complex* DFT also recognizes negative time and negative frequencies. Complex mathematics accommodates these concepts, although imaginary time and frequency have only a theoretical existence so far. *Complex* DFT is a symmetrical and mathematically comprehensive processing technology because it doesn't discriminate between negative and

Fig. 6 shows how the forward *complex* DFT algorithm works in the case of a complex timedomain signal. *xR*(*n*) is a real time domain signal whose frequency spectrum has an even real part and an odd imaginary part; conversely, the frequency spectrum of the imaginary part of the time domain signal *xI*(*n*) has an odd real part and an even imaginary part. However, as can be seen in Fig. 6, the actual frequency spectrum is the sum of the two individuallycalculated spectra. In reality, these two time domain signals are processed simultaneously,

> **Frequency Domain** *X* (*k*)*=XR* (*k*)+*XI* (*k*)

The imaginary part of the time-domain complex signal can be omitted and the time domain then becomes totally real, as is assumed in the numerical example shown in Fig. 7. A real sinusoidal signal with amplitude *M,* represented in a complex form, contains a positive ǚ0 and a negative frequency -ǚ0. The complex spectrum *X*(*k*) describes the signal in the

*Real Frequency Spectrum*  (odd)

*Imaginary Frequency Spectrum*  (even)

*Complex DFT*

**Time Domain**  *x(n)= xR* (*n*) + *j xI* (*n*) *xR* (*n*) *xI* (*n*) *Real time signal Imaginary time signal* 

*x*(0) to *x*(*N*-1) by a sinusoid and sums the results over the complete time-period.

signal. It is clear that *complex* DFT works with finite-length data.

which is the whole point of the Fast Fourier Transform (FFT) algorithm.

positive frequencies.

*Real Frequency Spectrum*  (even)

*Imaginary Frequency Spectrum*  (odd)

Fig. 6. Forward *complex* DFT algorithm

frequency domain. The frequency range of its real, Re *X*(*k*), and imaginary part, Im *X*(*k*), comprises both positive and negative frequencies simultaneously. Since the considered time domain signal is real, Re *X*(*k*) is even (the spectral values *A* and *B* have the same sign), while the imaginary part Im *X*(*k*) is odd (*C* is negative, *D* is positive).

The amplitude of each of the four spectral peaks is *M*/2, which is half the amplitude of the time domain signal. The single frequency interval under consideration [-¼ǚs÷¼ǚs] ([-0.5÷0.5] when normalized frequency is used) is symmetric with respect to a frequency of zero. The real frequency spectrum Re *X*(*k*) is used to reconstruct a cosine time domain signal, whilst the imaginary spectrum Im *X*(*k*) results in a negative sine wave, both with amplitude *M* in accordance with the complex analysis equation (Table 1). In a way analogous to the example shown in Fig. 7, a complex frequency spectrum can also be derived.

Fig. 7. Inverse *complex* DFT - reconstruction of a real time domain signal

Why is *complex* DFT used since it involves intricate complex number math? *Complex* DFT has persuasive advantages over *real* DFT and is considered to be the more comprehensive version. *Real* DFT is mathematically simpler and offers practical solutions to real world problems; by extension, negative frequencies are disregarded. Negative frequencies are always encountered in conjunction with complex numbers.

Complex Digital Signal Processing in Telecommunications 13

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

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 filter coefficient T, responsible for the central frequency, is updated by applying an

derivative of complex BP filter output *y*(*n*) with respect to the coefficient, which is subject to

E

In order to ensure the stability of the adaptive algorithm, the range of the step size P should

<sup>2</sup> <sup>0</sup> *<sup>P</sup> N* P

depends on the statistical characteristics of the input signal. In most practical situations, *P* is

V

E

( 1) ( ) Re[ ( ) ( )] *n n eny n* c . (11)

**+**

*eR*(*n*)

*yI*(*n*) *yR*(*n*)

*eI*(*n*)

(*n*) and *P* is a constant which

**+**

. (11)

(*n*) is the

 TP

The step sizeP controls the speed of convergence, ( ) denotes complex-conjugate, *y*<sup>c</sup>

Adaptive Complex Filter

z-1 sinT

z-1

bandwidth over a wide frequency range.

*xR*(*n*)

*xI*(*n*)

be set according to (Douglas, 1999):

approximately equal to 0.1.

adaptation.

The cost-function is the power of the BP/BS filter's output signal.

adaptive algorithm, for example LMS (Least Mean Square):

T

sinT

cosT

cosT

Adaptive Algoritm

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

where *N* is the filter order, ǔ2 is the power of the signal *y*<sup>c</sup>

*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* DFT.

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 the rest of the samples.

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

*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 transforms in particular.
