The electrical engineering handbook CH014

The electrical engineering handbook

Jenkins, W.K., Poularikas, A.D., Bomar, B.W., Smith, L.M., Cadzow, J.A “Digital Signal Processing”

The Electrical Engineering Handbook

Ed Richard C Dorf

Boca Raton: CRC Press LLC, 2000

© 2000 by CRC Press LLC

14 Digital Signal Processing

14.1 Fourier Transforms

Introduction • The Classical Fourier Transform for CT Signals • Fourier Series Representation of CT Periodic Signals • Generalized Complex Fourier Transform • DT Fourier Transform • Relationship between the CT and DT Spectra • Discrete Fourier Transform

14.2 Fourier Transforms and the Fast Fourier Transform

The Discrete Time Fourier Transform (DTFT) • Relationship to the Z-Transform • Properties • Fourier Transforms of Finite Time Sequences • Frequency Response of LTI Discrete Systems • The Discrete Fourier Transform • Properties of the DFT • Relation between DFT and Fourier Transform • Power, Amplitude, and Phase Spectra • Observations • Data Windowing • Fast Fourier Transform • Computation of the Inverse DFT

14.3 Design and Implementation of Digital Filters

Finite Impulse Response Filter Design • Infinite Impulse Response Filter Design • Finite Impulse Response Filter Implementation • Infinite Impulse Response Filter Implementation

14.4 Signal Restoration

Introduction • Attribute Sets: Closed Subspaces • Attribute Sets: Closed Convex Sets • Closed Projection Operators • Algebraic Properties of Matrices • Structural Properties of Matrices • Nonnegative Sequence Approximation • Exponential Signals and the Data Matrix • Recursive Modeling of Data

< t < ¥ A more recently developed set of discrete Fourier methods, including the discrete-time (DT) Fouriertransform and the discrete Fourier transform (DFT), are extensions of basic Fourier concepts for DT signalsand systems A DT signal is defined only for integer values of n in the range –¥ < n < ¥ The class of DT

Fourier methods is particularly useful as a basis for digital signal processing (DSP) because it extends the theory

of classical Fourier analysis to DT signals and leads to many effective algorithms that can be directly implemented

on general computers or special-purpose DSP devices

The Classical Fourier Transform for CT Signals

A CT signal s(t) and its Fourier transform S(jw) form a transform pair that are related by Eqs (14.1) for any

s(t) for which the integral (14.1a) converges:

(14.1a)

(14.1b)

In most literature Eq (14.1a) is simply called the Fourier transform, whereas Eq (14.1b) is called the Fourier

integral The relationship S(jw) = F {s(t)} denotes the Fourier transformation of s(t), where F { } is a symbolic

notation for the integral operator and where w is the continuous frequency variable expressed in radians per

second A transform pair s(t) «S(jw) represents a one-to-one invertible mapping as long as s(t) satisfies

conditions which guarantee that the Fourier integral converges

In the following discussion the symbol d(t) is used to denote a CT impulse function that is defined to be

zero for all t¹ 0, undefined for t = 0, and has unit area when integrated over the range –¥ < t < ¥ From Eq

(14.1a) it is found that F {d(t – t o)} = e–jwto due to the well-known sifting property of d(t) Similarly, from Eq

(14.1b) we find that F–1

{2pd(w – wo)} = ejwot, so that d(t – t o) «e–jwtoand e jwot« 2pd(w – wo) are Fouriertransform pairs By using these relationships, it is easy to establish the Fourier transforms of cos(wo t) and

sin(wo t), as well as many other useful waveforms, many of which are listed in Table 14.1

The CT Fourier transform is useful in the analysis and design of CT systems, i.e., systems that process CT

signals Fourier analysis is particularly applicable to the design of CT filters which are characterized by Fourier

magnitude and phase spectra, i.e., by |H(jw)| and arg H(jw), where H(jw) is commonly called the frequency

response of the filter

Properties of the CT Fourier Transform

The CT Fourier transform has many properties that make it useful for the analysis and design of linear CT

systems Some of the more useful properties are summarized in this section, while a more complete list of the

CT Fourier transform properties is given in Table 14.2 Proofs of these properties are found in Oppenheim

et al [1983] and Bracewell [1986] Note that F{ } denotes the Fourier transform operation, F –1{ } denotes

the inverse Fourier transform operation, and “*” denotes the convolution operation defined as

1 Linearity (superposition): F{af1(t) + bf2(t)} = aF{f1(t)} + bF{f2(t)}

(a and b, complex constants)

3 Frequency Shifting: e j wot f (t) = F–1{F(j(w – w o))}

s t S j ( ) = ( ) ( ) ej td

-¥

¥ò

f t f t f t1( ) * 2( ) = 1( ) ( ) - t f t dt2

-¥

¥ò

t

( )

ìíî

üýþ

= ( ) ( ) + ( ) ( )

¥

ò–

The above properties are particularly useful in CT system analysis and design, especially when the systemcharacteristics are easily specified in the frequency domain, as in linear filtering Note that Properties 1, 6, and

7 are useful for solving differential or integral equations Property 4 (time-domain convolution) provides the

Signal Fourier Transform Fourier Series Coefficients (if periodic)

=

= , otherwise cos w0t p d w w d w w[ ( - 0)+ ( + 0) ]

a a

a k

1 1 1 2 0

w p

0 1 0 1 0 1T k T k T

k

sinc æ è

ö ø

k

æ è

-ö ø

æ è

ö ø

1 0

, ,

basis for many signal-processing algorithms, since many systems can be specified directly by their impulse orfrequency response Property 3 (frequency shifting) is useful for analyzing the performance of communicationsystems where different modulation formats are commonly used to shift spectral energy among differentfrequency bands.

Fourier Spectrum of a CT Sampled Signal

The operation of uniformly sampling a CT signal s(t) at every T seconds is characterized by Eq (14.2), where d(t) is the CT impulse function defined earlier:

(a) f (t) is even

(b) f (t) is odd Negative t

Scaling:

(a) Time (b) Magnitude Differentiation

Integration Time shifting Modulation

w

w w

( )= ( )

( )+ ( ) [ ] = ( )+ ( )

-¥

-¥

-¥

¥

ò ò

1 2

2

2

0 0

cos

sin

F f( )-t =F*( )jw

F F

f at

a F

j a

af t aF j

è

ö ø

n

( )

é ë

cos sin

F

-¥

¥

( ) ( ) [ ] = ò ( ) ( - )

s t s ta( ) = a( ) ( ) t - nT s nT = a( ) ( ) t - nT

Since s a (t) is a CT signal, it is appropriate to apply the CT Fourier transform to obtain an expression for the

spectrum of the sampled signal:

(14.3)

Since the expression on the right-hand side of Eq (14.3) is a function of e j wT

, it is customary to express the

transform as F(e j wT ) = F{s a (t)} It will be shown later that if w is replaced with a normalized frequency w¢ = w/T, so that –p < w¢ < p, then the right side of Eq (14.3) becomes identical to the DT Fourier transform that

is defined directly for the sequence s[n] = s a (nT).

Fourier Series Representation of CT Periodic Signals

The classical Fourier series representation of a periodic time domain signal s(t) involves an expansion of s(t)

into an infinite series of terms that consist of sinusoidal basis functions, each weighted by a complex constant(Fourier coefficient) that provides the proper contribution of that frequency component to the complete

waveform The conditions under which a periodic signal s(t) can be expanded in a Fourier series are known

as the Dirichlet conditions They require that in each period s(t) has a finite number of discontinuities, a finite number of maxima and minima, and that s(t) satisfies the absolute convergence criterion of Eq (14.4) [Van

Valkenburg, 1974]:

(14.4)

It is assumed throughout the following discussion that the Dirichlet conditions are satisfied by all functionsthat will be represented by a Fourier series

The Exponential Fourier Series

If s(t) is a CT periodic signal with period T, then the exponential Fourier series expansion of s(t) is given by

(14.5a)where wo = 2p/T and where the an terms are the complex Fourier coefficients given by

(14.5b)

For every value of t where s(t) is continuous the right side of Eq (14.5a) converges to s(t) At values of t where s(t) has a finite jump discontinuity, the right side of Eq (14.5a) converges to the average of s(t–) and s(t+), where

For example, the Fourier series expansion of the sawtooth waveform illustrated in Fig 14.1 is characterized

by T = 2p, wo = 1, a0 = 0, and a n = a –n = A cos(np)/(jnp) for n = 1, 2, … The coefficients of the exponential Fourier series given by Eq (14.5b) can be interpreted as a spectral representation of s(t), since the a nth coefficient

represents the contribution of the (nw o )th frequency component to the complete waveform Since the a n termsare complex valued, the Fourier domain (spectral) representation has both magnitude and phase spectra For

example, the magnitude of the a n values is plotted in Fig 14.2 for the sawtooth waveform of Fig 14.1 The fact

that the a n terms constitute a discrete set is consistent with the fact that a periodic signal has a line spectrum;

n

a

j T n n

( )

-î

ü ý ï þ

i.e., the spectrum contains only integer multiples of the fundamental frequency wo Therefore, the equationpair given by Eq (14.5a) and (14.5b) can be interpreted as a transform pair that is similar to the CT Fouriertransform for periodic signals This leads to the observation that the classical Fourier series can be interpreted

as a special transform that provides a one-to-one invertible mapping between the discrete-spectral domain andthe CT domain

Trigonometric Fourier Series

Although the complex form of the Fourier series expansion is useful for complex periodic signals, the Fourierseries can be more easily expressed in terms of real-valued sine and cosine functions for real-valued periodic

signals In the following discussion it will be assumed that the signal s(t) is real valued for the sake of simplifying the discussion When s(t) is periodic and real valued it is convenient to replace the complex exponential form

of the Fourier series with a trigonometric expansion that contains sin(wo t) and cos(wo t) terms with

corre-sponding real-valued coefficients [Van Valkenburg, 1974] The trigonometric form of the Fourier series for a

real-valued signal s(t) is given by

(14.6a)where wo = 2p/T The bn and c n terms are real-valued Fourier coefficients determined by

and

(14.6b)

n

n n

2 2

An arbitrary real-valued signal s(t) can be expressed as a sum of even and odd components, s(t) = seven(t) +

sodd(t), where seven(t) = seven(–t) and sodd(t) = –sodd(–t), and where seven(t) = [s(t) + s(–t)]/2 and sodd(t) = [s(t) – s(–t)]/2 For the trigonometric Fourier series, it can be shown that seven(t) is represented by the (even) cosine terms in the infinite series, sodd(t) is represented by the (odd) sine terms, and b0 is the dc level of the signal.Therefore, if it can be determined by inspection that a signal has a dc level, or if it is even or odd, then thecorrect form of the trigonometric series can be chosen to simplify the analysis For example, it is easily seenthat the signal shown in Fig 14.3 is an even signal with a zero dc level Therefore, it can be accurately represented

by the cosine series with b n = 2A sin(pn/2)/(pn/2), n = 1, 2, …, as illustrated in Fig 14.4 In contrast, note thatthe sawtooth waveform used in the previous example is an odd signal with zero dc level, so that it can becompletely specified by the sine terms of the trigonometric series This result can be demonstrated by pairing

each positive frequency component from the exponential series with its conjugate partner; i.e., c n = sin(nw o t)

= a n e jn wot + a –n e –jn wot , whereby it is found that c n = 2A cos(np)/(np) for this example In general, it is found that a n = (b n – jc n )/2 for n = 1, 2, …, a0 = b0, and a –n = a n* The trigonometric Fourier series is common in thesignal processing literature because it replaces complex coefficients with real ones and often results in a simplerand more intuitive interpretation of the results

Convergence of the Fourier Series

The Fourier series representation of a periodic signal is an approximation that exhibits mean-squared

conver-gence to the true signal If s(t) is a periodic signal of period T and s¢(t) denotes the Fourier series approximation

of s(t), then s(t) and s¢(t) are equal in the mean-squared sense if

(14.7)

Even when Eq (14.7) is satisfied, mean-squared error (mse) convergence does not guarantee that s(t) = s¢(t)

at every value of t In particular, it is known that at values of t where s(t) is discontinuous the Fourier series converges to the average of the limiting values to the left and right of the discontinuity For example, if t0 is a

point of discontinuity, then s¢(t0) = [s(t0) + s(t+0)]/2, where s(t0) and s(t0+) were defined previously (note that

at points of continuity, this condition is also satisfied by the very definition of continuity) Since the Dirichlet

conditions require that s(t) have at most a finite number of points of discontinuity in one period, the set S t

such that s(t) ¹ s¢(t) within one period contains a finite number of points, and S t is a set of measure zero in

the formal mathematical sense Therefore, s(t) and its Fourier series expansion s¢(t ) are equal almost everywhere, and s(t) can be considered identical to s¢(t) for analysis in most practical engineering problems.

FIGURE 14.3 Periodic CT signal used in Fourier series example 2.

0

The condition described above of convergence almost

everywhere is satisfied only in the limit as an infinite number

of terms are included in the Fourier series expansion If the

infinite series expansion of the Fourier series is truncated to

a finite number of terms, as it must always be in practical

applications, then the approximation will exhibit an

oscilla-tory behavior around the discontinuity, known as the Gibbs

phenomenon [Van Valkenburg, 1974] Let s N ¢ (t) denote a

truncated Fourier series approximation of s(t), where only

the terms in Eq (14.5a) from n = –N to n = N are included

if the complex Fourier series representation is used or where

only the terms in Eq (14.6a) from n = 0 to n = N are included

if the trigonometric form of the Fourier series is used It is

well known that in the vicinity of a discontinuity at t0 the

Gibbs phenomenon causes s N ¢ (t) to be a poor approximation

to s(t) The peak magnitude of the Gibbs oscillation is 13% of the size of the jump discontinuity s (t0) – s(t+0)

regardless of the number of terms used in the approximation As N increases, the region which contains the oscillation becomes more concentrated in the neighborhood of the discontinuity, until, in the limit as N approaches infinity, the Gibbs oscillation is squeezed into a single point of mismatch at t0 The Gibbs phenom-enon is illustrated in Fig 14.5, where an ideal low-pass frequency response is approximated by an impulse

response function that has been limited to having only N nonzero coefficients, and hence the Fourier series

expansion contains only a finite number of terms

If s¢(t) in Eq (14.7) is replaced by s N ¢ (t) it is important to understand the behavior of the error mse N as a

increased Therefore, when applying Fourier series analysis, including more terms always improves the accuracy

of the signal representation

Fourier Transform of Periodic CT Signals

For a periodic signal s(t) the CT Fourier transform can then be applied to the Fourier series expansion of s(t)

to produce a mathematical expression for the “line spectrum” that is characteristic of periodic signals:

(14.10)

The spectrum is shown in Fig 14.6 Note the similarity between the spectral representation of Fig 14.6 and theplot of the Fourier coefficients in Fig 14.2, which was heuristically interpreted as a line spectrum Figures 14.2 and

FIGURE 14.5 Gibbs phenomenon in a low-pass

digital filter caused by truncating the impulse

14.6 are different, but equivalent, representations of the Fourier line spectrum that is characteristic of periodicsignals.

Generalized Complex Fourier Transform

The CT Fourier transform characterized by Eqs (14.11a) and (14.11b) can be generalized by considering the

variable jw to be the special case of u = s + jw with s = 0, writing Eqs (14.11) in terms of u, and interpreting

u as a complex frequency variable The resulting complex Fourier transform pair is given by Eqs (14.11a) and

DT Fourier Transform

The DT Fourier transform (DTFT) is obtained directly in terms of the sequence samples s[n] by taking the relationship obtained in Eq (14.3) to be the definition of the DTFT By letting T = 1 so that the sampling

period is removed from the equations and the frequency variable is replaced with a normalized frequency w¢

= wT, the DTFT pair is defined by Eqs (14.12) In order to simplify notation it is not customary to distinguishbetween w and w¢, but rather to rely on the context of the discussion to determine whether w refers to the

normalized (T = 1) or to the unnormalized (T ¹ 1) frequency variable.

1 2p

s s

p p

The spectrum S(e jw¢) is periodic in w¢ with period 2p The fundamental period in the range –p < w¢ £ p,sometimes referred to as the baseband, is the useful frequency range of the DT system because frequencycomponents in this range can be represented unambiguously in sampled form (without aliasing error) In much

of the signal-processing literature the explicit primed notation is omitted from the frequency variable However,the explicit primed notation will be used throughout this section because there is a potential for confusionwhen so many related Fourier concepts are discussed within the same framework

By comparing Eqs (14.3) and (14.12a), and noting that w¢ = wT, we see that

(14.13)

where s[n] = s(t)| t = nT This demonstrates that the spectrum of s a (t) as calculated by the CT Fourier transform

is identical to the spectrum of s[n] as calculated by the DTFT Therefore, although s a (t) and s[n] are quite

different sampling models, they are equivalent in the sense that they have the same Fourier domain tation A list of common DTFT pairs is presented in Table 14.3 Just as the CT Fourier transform is useful in

represen-CT signal system analysis and design, the DTFT is equally useful for DT system analysis and design

Sequence Fourier Transform

sin sin

w w +

1 0

,

x n[ ] = ìí £n £ Mî

1 0 0

, , otherwise

sin sin

w w

In the same way that the CT Fourier transform was found to be a special case of the complex Fourier

transform (or bilateral Laplace transform), the DTFT is a special case of the bilateral z-transform with z = e j w¢t

The more general bilateral z-transform is given by

Properties of the DTFT

Since the DTFT is a close relative of the classical CT Fourier transform, it should come as no surprise thatmany properties of the DTFT are similar to those of the CT Fourier transform In fact, for many of the propertiespresented earlier there is an analogous property for the DTFT The following list parallels the list that waspresented in the previous section for the CT Fourier transform, to the extent that the same property exists Amore complete list of DTFT pairs is given in Table 14.4:

1 Linearity (superposition): DTFT{af1[n] + bf2[n]} = aDTFT{f1[n]} + bDTFT{f2[n]}

(a and b, complex constants)

2 Index Shifting: DTFT{f [n – n o ]} = e –j wno DTFT{f [n]}

3 Frequency Shifting: e j won f [n] = DTFT–1{F( j(w – w o))}

it allows engineers to work with the frequency response of the system in order to achieve proper shaping ofthe input spectrum, or to achieve frequency selective filtering for noise reduction or signal detection Also,Property 3 (frequency shifting) is useful for the analysis of modulation and filtering common in both analogand digital communication systems

Relationship between the CT and DT Spectra

Since DT signals often originate by sampling a CT signal, it is important to develop the relationship betweenthe original spectrum of the CT signal and the spectrum of the DT signal that results First, the CT Fouriertransform is applied to the CT sampling model, and the properties are used to produce the following result:

= ( ) ( ) ì í ï ( )

-î

ü ý ï þ

In this section it is important to distinguish between w and w¢, so the explicit primed notation is used in thefollowing discussion where needed for clarification Since the sampling function (summation of shifted

impulses) on the right-hand side of the above equation is periodic with period T it can be replaced with a CT

Fourier series expansion as follows:

Applying the frequency-domain convolution property of the CT Fourier transform yields

(14.16a)where ws = (2p/T) is the sampling frequency (rad/s) An alternate form for the expression of Eq (14.16a) is

(14.16b)

Sequence Fourier Transform

x[n] X(e jw )

y[n] Y(e jw ) 1.

( )

= -¥

¥å

n

s n

where w¢ = wT is the normalized DT frequency axis expressed in radians Note that S(ej wT ) = S(e jw¢) consists

of an infinite number of replicas of the CT spectrum S( jw), positioned at intervals of (2p/T) on the w axis (or

at intervals of 2p on the w¢ axis), as illustrated in Fig 14.7 If S( jw) is band limited with a bandwidth wc and

if T is chosen sufficiently small so that w s > 2wc , then the DT spectrum is a copy of S( jw) (scaled by 1/T) in

the baseband The limiting case of ws = 2wc is called the Nyquist sampling frequency Whenever a CT signal

is sampled at or above the Nyquist rate, no aliasing distortion occurs (i.e., the baseband spectrum does notoverlap with the higher-order replicas) and the CT signal can be exactly recovered from its samples by extracting

the baseband spectrum of S(e jw¢) with an ideal low-pass filter that recovers the original CT spectrum by removing

all spectral replicas outside the baseband and scaling the baseband by a factor of T.

Discrete Fourier Transform

To obtain the DFT the continuous-frequency domain of the DTFT is sampled at N points uniformly spaced around the unit circle in the z-plane, i.e., at the points w k = (2pk/N), k = 0, 1, …, N – 1 The result is the DFT

transform pair defined by Eqs (14.17a) and (14.17b) The signal s[n] is either a finite-length sequence of length

N or it is a periodic sequence with period N.

Regardless of whether s[n] is a finite-length or a periodic sequence, the DFT treats the N samples of s[n] as

though they characterize one period of a periodic sequence This is an important feature of the DFT, and onethat must be handled properly in signal processing to prevent the introduction of artifacts Important properties

of the DFT are summarized in Table 14.5 The notation (k) N denotes k modulo N, and R N [n] is a rectangular window such that R N [n] = 1 for n = 0, …, N – 1, and R N [n] = 0 for n < 0 and n ³ N The transform relationship given by Eqs (14.17a) and (14.17b) is also valid when s[n] and S[k] are periodic sequences, each of period N.

In this case, n and k are permitted to range over the complete set of real integers, and S[k] is referred to as the

discrete Fourier series (DFS) The DFS is developed by some authors as a distinct transform pair in its ownright [Oppenheim and Schafer, 1975] Whether or not the DFT and the DFS are considered identical or distinct

is not very important in this discussion The important point to be emphasized here is that the DFT treats s[n]

as though it were a single period of a periodic sequence, and all signal processing done with the DFT willinherit the consequences of this assumed periodicity

Properties of the DFT

Most of the properties listed in Table 14.5 for the DFT are similar to those of the z-transform and the DTFT,

although there are some important differences For example, Property 5 (time-shifting property), holds for

circular shifts of the finite-length sequence s[n], which is consistent with the notion that the DFT treats s[n]

as one period of a periodic sequence Also, the multiplication of two DFTs results in the circular convolution

S k s n e k N

j kn N n

N

j kn N n

= -

å å

2

0 1

2

0 1

TABLE 14.5 Properties of the Discrete Fourier Transform (DFT)

Finite-Length Sequence (Length N) N-Point DFT (Length N)

1 2

1 2

î

ï ï ï ï ï

of the corresponding DT sequences, as specified by Property 7 This latter property is quite different from the

linear convolution property of the DTFT Circular convolution is simply a linear convolution of the periodic extensions of the finite sequences being convolved, where each of the finite sequences of length N defines the

structure of one period of the periodic extensions

For example, suppose it is desired to implement the following finite impulse response (FIR) digital filter,

(14.18)

the output of which is obtained by transforming h[n] and s[n] into H[k] and S[k] via the DFT (FFT), multiplying the transforms point-wise to obtain Y[k] = H[k]S[k], and then using the inverse DFT (FFT) to obtain y[n] = IDFT{Y[k]} If s[n] is a finite sequence of length M, then the result of the circular convolution implemented

by the DFT will correspond to the desired linear convolution if and only if the block length of the DFT is

chosen so that NDFT ³ N + M – 1 and both h[n] and s[n] are padded with zeros to form blocks of length NDFT

Relationships among Fourier Transforms

Figure 14.8 illustrates the functional relationships among the various forms of CT and DT Fourier transformsthat have been discussed in the previous sections The family of CT is shown on the left side of Fig 14.8, whereasthe right side of the figure shows the hierarchy of DTFTs Fourier transforms The complex Fourier transform

is identical to the bilateral Laplace transform, and it is at this level that the classical Laplace transform techniquesand the Fourier transform techniques become identical

-Defining Terms

Continuous-time (CT) impulse function: A generalized function d(t) defined to be zero for all t ¹ 0,

undefined at t = 0, and having the special property that

Circular convolution: A convolution of finite-length sequences in which the shifting operation is performed

circularly within the finite support interval Alternatively called periodic convolution.

Dirichlet conditions: Conditions that must be satisfied in order to expand a periodic signal s(t) in a Fourier series: each period of s(t) must have a finite number of discontinuities, a finite number of maxima and

minima, and must be satisfied, where T is the period.

Gibbs phenomenon: Oscillatory behavior of Fourier series approximations in the vicinity of finite jumpdiscontinuities

Line spectrum: A common term for Fourier transforms of periodic signals for which the spectrum has nonzero

components only at integer multiples of the fundamental frequency

Mean-squared error (mse): A measure of “closeness” between two functions given by

where T is the period.

Nyquist sampling frequency: Minimum sampling frequency for which a CT signal s(t) can be perfectly reconstructed from a set of uniformly spaced samples s(nT).

Orthonormal set: A countable set of functions for which every pair in the set is mathematically orthogonalaccording to a valid norm, and for which each element of the set has unit length according to the samenorm The Fourier basis functions form an orthonormal set according to the mse norm

Trigonometric expansion: A Fourier series expansion for a real-valued signal in which the basis functions

are chosen to be sin(nw o t) and cos(nwo t)

Related Topic

16.1 Spectral Analysis

References

R N Bracewell, The Fourier Transform, 2nd ed., New York: McGraw-Hill, 1986.

W K Jenkins, “Fourier series, Fourier transforms, and the discrete Fourier transform,” in The Circuits and Filters Handbook, Chen, (ed.), Boca Raton, Fla.: CRC Press, 1995.

A V Oppenheim, A S Willsky, and I T Young, Signals and Systems, Englewood Cliffs, N.J.: Prentice-Hall, 1983.

A V Oppenheim and R W Schafer, Discrete-Time Signal Processing, Englewood Cliffs, N.J.: Prentice-Hall, 1989.

A V Oppenheim and R W Schafer, Digital Signal Processing, Englewood Cliffs, N.J.: Prentice-Hall, 1975.

M E., VanValkenburg, Network Analysis, Englewood Cliffs, N.J.: Prentice-Hall, 1974.

Carmel, Ind.: SAMS Publishing Co., 1993

A classic reference on the CT Fourier transform is Bracewell [1986]

2

T

14.2 Fourier Transforms and the Fast Fourier Transform

Alexander D Poularikas

The Discrete Time Fourier Transform (DTFT)

The discrete time Fourier transform of a signal {f(n)} is defined by

(14.19)and its inverse discrete time Fourier transform (IDTFT) is give by

Table 14.6 tabulates the DTFT properties of discrete time sequences

Fourier Transforms of Finite Time Sequences

The trancated Fourier transform of a sequence is given by

–

f n F ( ) = 2p 1 ò ( ) w ej nd w

w p

j n n

( ) =

-æ è

ö ø

where w(n) is a window function that extends from n = 0 to n = N – 1 If the value of the sequence is unity for all n’s, the window is known as the rectangular one From (14.23) we observe that the trancation of a

sequence results in a smoothered version of the exact spectrum

Frequency Response of LTI Discrete Systems

A first order LTI discrete system is described by the difference equation

y(n) + a1y(n – 1) = b0x(n) + b1x(n – 1)

The DTFT of the above equation is given by

Y( w) + a1e–jwY( w) = b0X( w) + b1e–jw X( w)from which we write the system function

To approximate the continuous time Fourier transform using the DTFT we follow the following steps:

1 Select the time interval T such that F(w c) » 0 for all *wc* > p/T wc designates the frequency of a continuoustime function

2 Sample f (t) at times nT to obtain f (nT).

3 Compute the DFT using the sequence {Tf(nT)}.

4 The resulting approximation is then F(wc) » F(w) for –p/T < wc < p/T.

The Disrete Fourier Transform

One of the methods, and one that is used extensively, calls for replacing continuous Fourier transforms by an

equivalent discrete Fourier transform (DFT) and then evaluating the DFT using the discrete data However,

evaluating a DFT with 512 samples (a small number in most cases) requires more than 1.5 ´ 106 mathematical

operations It was the development of the fast Fourier transform (FFT), a computational technique that reduces

1 2

w

w w( ) = ( )

( ) =

+ +

-

-0 1 1

1

the number of mathematical operations in the evaluation of the DFT to N log2 (N) (approximately 2.5 ´10

operations for the 512-point case mentioned above), that makes DFT an extremely useful tool in most all fields

of science and engineering

A data sequence is available only with a finite time window from n = 0 to n = N – 1 The transform is discretized for N values by taking samples at the frequencies 2p/NT, where T is the time interval between sample points Hence, we define the DFT of a sequence of N samples for 0 £ k £ N – 1 by the relation

(14.24)

where N = number of sample values, T = sampling time interval, (N – 1)T = signal length, f (nT ) = sampled form

of f (t) at points nT, W = (2p/T)1/N = w s /N = frequency sampling interval, e –iWT = Nth principal root of unity, and j = The inverse DFT is given by

N

j Tnk n

-= -

å å

0 1

= -

å å

0 1

0 1

N

N nk n

= -

å å

0 1

N

N nk k

-= -

å å

0 1

0 1

As a consequence, the sequences f(n) and F(k) as defined by (14.26) and (14.27) are also N-periodic.

It is generally convenient to adopt the convention

to represent the transform pair (14.26) and (14.27)

Properties of the DFT

A detailed discussion of the properties of DFT can be found in the cited references at the end of this section

In what follows we consider a few of these properties that are of value for the development of the FFT

1 Linearity:

2 Complex conjugate: If f (n) is real, N/2 is an integer and { f(n)} « { F(k)}, then

ö ø

-æ è ç

ö ø

Verify Parseval’s theorem for the sequence {f (n)} = {1, 2, –1, 3}.

Similarly, we find

Introducing these values in (14.39) we obtain

12 + 22 + (–1)2 + 32 = 1/4[52 + (2 + j )(2 – j ) + 52 +(2 – j )(2 + j )] or 15 = 60/4which is an identity, as it should have been

Relation between DFT and Fourier Transform

The sampled form of a continuous function f(t) can be represented by N equally spaced sampled values f(n)

such that

where T is the sampling interval The length of the continuous function is L = NT, where f (N) = f (0).

We denote the sampled version of f(t) by f s (t), which may be represented by a sequence of impulses.

Mathematically it is represented by the expression

ü ý ï þ

-f n

k N

n

N

0 1

=

-å å

Taking the Fourier transform of f s (t) in (14.41) we obtain

(14.42)

Equation (14.42) yields F s(w) for all values of w However, if we are only interested in the values of Fs(w) at aset of discrete equidistant points, then (14.42) is expressed in the form [see also (14.24)]

(14.43)

where W = 2p/L = 2p/NT Therefore, comparing (14.26) and (14.43) we observe that we can find F(w)from

F s(w) using the relation

(14.44)

Power, Amplitude, and Phase Spectra

If f(t) represents voltage or current waveform supplying a load of 1 W, the left-hand side of Parseval’s theorem

(14.39) represents the power dissipated in the 1-W resistor Therefore, the right-hand side represents the powercontributed by each harmonic of the spectrum Thus the DFT power spectrum is defined as

For real f(n) there are only (N/2 + 1) independent DFT spectral points as the complex conjugate property

shows (14.31) Hence we write

The amplitude spectrum is readily found from that of a power spectrum, and it is defined as

The power and amplitude spectra are invariant with respect to shifts of the data sequence { f(n)}.

The phase spectrum of a sequence {f (n)} is defined as

(14.48)

As in the case of the power spectrum, only (N/2 + 1) of the DFT phase spectral points are independent for real {f(n)} For a real sequence {f(n)} the power spectrum is an even function about the point k = N/2 and the phase spectrum is an odd function about the point k = N/2.

å ò

å ò å

0 1

0 1

0 1

1 The frequency spacing Dw between coefficients is

(14.49)

2 The reciprocal of the record length defines the frequency resolution

3 If the number of samples N is fixed and the sampling time is increased, the record length and the precision

of frequency resolution is increased When the sampling time is decreased, the opposite is true

4 If the record length is fixed and the sampling time is decreased (N increases), the resolution stays the same and the computed accuracy of F(nW) increases

5 If the record length is fixed and the sampling time is increased (N decreases), the resolution stays the same and the computed accuracy of F(nW) decreases

Data Windowing

To produce more accurate frequency spectra it is recommended that the data are weighted by a window function.

Hence, the new data set will be of the form{f(n) w(n)} The following are the most commonly used windows:

1 Triangle (Fejer, Bartlet) window:

ï ï î

ï ï

è

ö ø

é ë

ê ê

ù û

ú ú

K

m

è ç

ö ø

5 Blackman-Harris window Harris used a gradient search technique to find three- and four-term sion of (14.53) that either minimized the maximum sidelobe level for fixed mainlobe width, or tradedmainlobe width versus minimum sidelobe level (see Table 14.7)

expan-6 Centered Gaussian window:

(14.56)

Fast Fourier Transform

One of the approaches to speed the computation of the DFT of a sequence is the decimation-in-time method This approach is one of breaking the N-point transform into two (N/2)-point transforms, breaking each (N/2)- point transform into two (N/4)-point transforms, and continuing the above process until we obtain the two- point transform We start with the DFT expression and factor it into two DFTs of length N/2:

ö ø

÷ é

ë

ê ê

ù û

ö ø

÷ é

ë

ê ê ê

ù û

ú ú ú

= æ è ç

ö ø

F k f n W n

f n W n

N kn n

N

N kn n

=

=

even odd

0 2

1 1

Letting n = 2m in the first sum and n = 2m + 1 in the second, (14.57) becomes

= W N/2 mk Since the DFT is periodic, F1(k) = F1(k + N/2) and F2(k) = F2(k + N/2).

We next apply the same procedure to each N/2 samples, where f11(m) = f1(2m) and f21(m) = f2(2m + 1), m = 0,1, , (N/4) – 1 Hence,

N

( )–

( ) ( )–

= /

N k m

N

N mk

1 0

2 1

2

2 0

point DFT of even-indexed sequence

point DFT of odd-indexed sequence

N

k N

N k

ö ø

N k m

N

N mk

0

4 1

4 2

21 0

N k

1 11

2 21

2 21

ö ø

Example 2

To find the FFT of the sequence {2, 3, 4, 5} we first bit reverse the position of the elements from their priority{00, 01, 10, 11} to {00, 10, 01, 11} position The new sequence is {2, 4, 3, 5} (see also Fig 14.9) Using (14.60)and (14.61) we obtain

From (14.61) the output is

F (0) = F1(0) + W4F2(0)

F (1) = F1(1) + W4F2(1)

F (2) = F1(0) – W4F2(0)

F (3) = F1(1) – W4F2(1)

Computation of the Inverse DFT

To find the inverse FFT using an FFT algorithm, we use the relation

1 2 0

1 2 0

0

1

2 1

1 2

0 1

1 2 1

0 2

2 2 0

2 2 0

1 2 0

1

2 1

4 1

2 0

2 1

4 1

4 1

For other transforms and their fast algorithms the reader should consult the references given at the end ofthis section.

Table 14.8 gives the FFT subroutine for fast implementation of the DFT of a finite sequence

SUBROUTINE FOUR1 (DATA, NN, ISIGN)

Replaces DATA by its discrete Fourier transform, if SIGN is input as 1; or replaces DATA by NN times its inverse discrete Fourier transform, if ISIGN is input as –1 DATA is a complex array of length NN or, equivalently, a real

array of length 2*NN NN must be an integer power of 2.

REAL*8 WR, WI, WPR, WPI, WTEMP, THETA Double precision for the trigonometric recurrences DIMENSION DATA (2*NN)

N=2*NN

J=1

DO 11 I=1, N, 2 This is the bit-reversal section of the routine.

IF (J.GT.I) THEN Exchange the two complex numbers TEMPR=DATA(J)

GO TO 1

ENDIF

J=J+M

11 CONTINUE

MMAX=2 Here begins the Danielson-Lanczos section of the routine.

2 IF (N.GT.MMAX) THEN Outer loop executed log 2 NN times.

TEMPI=SNGL(WR)*DATA(J+1)+SNGL(WI)*DATA(J) DATA(J)=DATA(I)-TEMPR

DATA(J+1)=DATA(I+1)-TEMPI DATA(I)=DATA(I)+TEMPR DATA(I+1)=DATA(I+1)+TEMPI

Source: ©1986 Numerical Recipes Software From Numerical Recipes: The Art of Scientific Computing, published by Cambridge

University Press Used by permission.

Defining Terms

FFT: A computational technique that reduces the number of mathematical operations in the evaluation of

the discrete Fourier transform (DFT) to N log2 N.

Phase spectrum: All phases associated with the spectrum harmonics.

Power spectrum: A power contributed by each harmonic of the spectrum

Window: Any appropriate function that multiplies the data with the intent to minimize the distortions ofthe Fourier spectra

Related Topic

14.1 Fourier Transforms

References

A Ahmed and K R Rao, Orthogonal Transforms for Digital Signal Processing, New York: Springer-Verlag, 1975.

E R Blahut, Fast Algorithms for Digital Signal Processing, Reading, Mass.: Addison-Wesley, 1987.

E O Bringham, The Fast Fourier Transform, Englewood Cliffs, N.J.: Prentice-Hall, 1974.

F D Elliot, Fast Transforms, Algorithms, Analysis, Applications, New York: Academic Press, 1982.

H J Nussbaumer, Fast Fourier Transform and Convolution Algorithms, New York: Springer-Verlag, 1982.

A D Poularikas and S Seely, Signals and System 2nd ed., Melbourne, FL: Krieger Publishing, 1995.

Further Information

A historical overview of the fast Fourier transform can be found in J.W Cooley, P.A.W Lewis, and P.D Welch,

“Historical notes on the fast Fourier transform,” IEEE Trans Audio Electroacoust., vol AV-15, pp 76–79, June

1967

Fast algorithms appear frequently in the monthly magazine Signal Processing, published by The Institute of

Electrical and Electronics Engineers

14.3 Design and Implementation of Digital Filters

Bruce W Bomar and L Montgomery Smith

A digital filter is a linear, shift-invariant system for computing a discrete output sequence from a discrete

input sequence The input/output relationship is defined by the convolution sum

where x(n) is the input sequence, y(n) is the output sequence, and h(n) is the impulse response of the filter The filter is often conveniently described in terms of its frequency characteristics that are given by the transfer function H( e jw) The impulse response and transfer function are a Fourier transform pair:

1

Closely related to the Fourier transform of h(n) is the z-transform defined by

The Fourier transform is then the z-transform evaluated on the unit circle in the z-plane (z = e jw) An important

property of the z-transform is that z–1 H(z) corresponds to h( n–1), so z–1 represents a one-sample delay, termed

a unit delay.

In this section, attention will be restricted to frequency-selective filters These filters are intended to pass

frequency components of the input sequence in a given band of the spectrum while blocking the rest Typical

frequency-selective filter types are low-pass, high-pass, bandpass, and band-reject Other special-purpose filters

exist, but their design is an advanced topic that will not be addressed here In addition, special attention is

given to causal filters, that is, those for which the impulse response is identically zero for negative n and thus

can be realized in real time Digital filters are further separated into two classes depending on whether theimpulse response contains a finite or infinite number of nonzero terms

Finite Impulse Response Filter Design

The objective of finite impulse response (FIR) filter design is to determine N + 1 coefficients

h(0), h(1), , h(N )

so that the transfer function H(e jw) approximates a desired frequency characteristic H d (e jw) All other impulse

response coefficients are zero An important property of FIR filters for practical applications is that they can

be designed to be linear phase; that is, the transfer function has the form

H(ejw) = A(ejw)e–jwN/2

where the amplitude A(e jw) is a real function of frequency The desired transfer function can be similarly written

Hd(ejw) = Ad(ejw)e–jwN/2

where A d (e jw) describes the amplitude of the desired frequency-selective characteristics For example, the

amplitude frequency characteristics of an ideal low-pass filter are given by

where wc is the cutoff frequency of the filter.

A linear phase characteristic ensures that a filter has a constant group delay independent of frequency Thus,all frequency components in the signal are delayed by the same amount, and the only signal distortionintroduced is that imposed by the filter’s frequency-selective characteristics Since a FIR filter can only approx-imate a desired frequency-selective characteristic, some measures of the accuracy of approximation are needed

to describe the quality of the design These are the passband ripple d p , the stopband attenuation d s, and the

transition bandwidth Dw These quantities are illustrated in Fig 14.10 for a prototype low-pass filter Thepassband ripple gives the maximum deviation from the desired amplitude (typically unity) in the region wherethe input signal spectral components are desired to be passed unattenuated The stopband attenuation givesthe maximum deviation from zero in the region where the input signal spectral components are desired to beblocked The transition bandwidth gives the width of the spectral region in which the frequency characteristics

for otherwise

* *

of the transfer function change from the passband to the stopband values Often, the passband ripple andstopband attenuation are specified in decibels, in which case their values are related to the quantities dp and ds by

FIR Filter Design by Windowing

The windowing design method is a computationally efficient technique for producing nonoptimal filters Filtersdesigned in this manner have equal passband ripple and stopband attenuation:

dp = ds = dThe method begins by finding the impulse response of the desired filter from

For ideal low-pass, high-pass, bandpass, and band-reject frequency-selective filters, the integral can be solved

in closed form The impulse response of the filter is then found by multiplying this ideal impulse response with

a window w(n) that is identically zero for n < 0 and for n > N:

Some commonly used windows are defined as follows:

1 Rectangular (truncation)

stopband attenuation ds, and transition bandwidth Dw.

2 Hamming

3 Kaiser

In general, windows that slowly taper the impulse response to zero result in lower passband ripple and a widertransition bandwidth Other windows (e.g., Hamming, Blackman) are also sometimes used but not as often asthose shown above

Of particular note is the Kaiser window where I0(.) is the 0th-order modified Bessel function of the first kindand b is a shape parameter The proper choice of N and b allows the designer to meet given passband ripple/stopband attenuation and transition bandwidth specifications Specifically, using S, the stopband atten-

uation in dB, the filter order must satisfy

Then, the required value of the shape parameter is given by

As an example of this design technique, consider a low-pass filter with a cutoff frequency of wc = 0.4p Theideal impulse response for this filter is given by

Choosing N = 8 and a Kaiser window with a shape parameter of b = 0.5 yields the following impulse response

coefficients:

h(0) = h(8) = –0.07568267 h(1) = h(7) = –0.06236596 h(2) = h(6) = 0.09354892

0

p for otherwise

ï ï î

ï ï

b

b

/

for otherwise

/ /

h(3) = h(5) = 0.30273070 h(4) = 0.40000000

Design of Optimal FIR Filters

The accepted standard criterion for the design of optimal FIR filters is to minimize the maximum value of theerror function

E( ejw) = Wd( ejw)*Ad( ejw) – A( ejw)*

over the full range of –p £ w £ p Wd (e jw) is a desired weighting function used to emphasize specifications in

a given frequency band The ratio of the deviation in any two bands is inversely proportional to the ratio oftheir respective weighting

A consequence of this optimization criterion is that the frequency characteristics of optimal filters are

equiripple: although the maximum deviation from the desired characteristic is minimized, it is reached several

times in each band Thus, the passband and stopband deviations oscillate about the desired values with equal

amplitude in each band Such approximations are frequently referred to as minimax or Chebyshev

approxima-tions In contrast, the maximum deviations occur near the band edges for filters designed by windowing

Equiripple FIR filters are usually designed using the Parks-McClellan computer program [Parks and Burrus, 1987], which uses the Remez exchange algorithm to determine iteratively the extremal frequencies at which the

maximum deviations in the error function occur A listing of this program along with a detailed description

of its use is available in several references including Parks and Burrus [1987] and DSP Committee [1979] Theprogram is executed by specifying as inputs the desired band edges, gain for each band (usually 0 or 1), bandweighting, and FIR length If the resulting filter has too much ripple in some bands, those bands can be weightedmore heavily and the filter redesigned Details on this design procedure are discussed in Rabiner [1973], alongwith approximate design relationships which aid in selecting the filter length needed to meet a given set ofspecifications

Although we have focused attention on the design of frequency-selective filters, other types of FIR filtersexist For example, the Parks-McClellan program will also design linear-phase FIR filters for differentiatingbroadband signals and for approximating the Hilbert transform of such signals A simple modification to thisprogram permits arbitrary magnitude responses to be approximated with linear-phase filters Other designtechniques are available that permit the design of FIR filters which approximate an arbitrary complex response[Parks and Burrus, 1987; Chen and Parks, 1987], and, in cases where a nonlinear phase response is acceptable,design techniques are available that give a shorter impulse response length than would be required by a linear-

phase design [Goldberg et al., 1981].

As an example of an equiripple filter design, an 8th-order low-pass filter with a passband 0 £ w £ 0.3p, astopband 0.5p £ w £ p, and equal weighting for each band was designed The impulse response coefficientsgenerated by the Parks-McClellan program were as follows:

h(0) = h(8) = –0.06367859 h(1) = h(7) = –0.06912276 h(2) = h(6) = 0.10104360 h(3) = h(5) = 0.28574990 h(4) = 0.41073000

These values can be compared to those for the similarly specified filter designed in the previous subsectionusing the windowing method

Infinite Impulse Response Filter Design

An infinite impulse response (IIR) digital filter requires less computation to implement than a FIR digital

filter with a corresponding frequency response However, IIR filters cannot generally achieve a perfect phase response and are more susceptible to finite wordlength effects

linear-Techniques for the design of IIR analog filters are well established For this reason, the most important class

of IIR digital filter design techniques is based on forcing a digital filter to behave like a reference analog filter

This can be done in several different ways For example, if the analog filter impulse response is h a (t) and the digital filter impulse response is h(n), then it is possible to make h(n) = h a (nT), where T is the sample spacing

of the digital filter Such designs are referred to as impulse-invariant [Parks and Burrus, 1987] Likewise, if g a (t)

is the unit step response of the analog filter and g(n) is the unit step response of the digital filter, it is possible

to make g(n) = g a (nT), which gives a step-invariant design [Parks and Burrus, 1987].

The step-invariant and impulse-invariant techniques perform a time domain matching of the analog anddigital filters but can produce aliasing in the frequency domain For frequency-selective filters it is better toattempt matching frequency responses This task is complicated by the fact that the analog filter response isdefined for an infinite range of frequencies (W = 0 to ¥), while the digital filter response is defined for a finiterange of frequencies (w = 0 to p) Therefore, a method for mapping the infinite range of analog frequencies

W into the finite range from w = 0 to p, termed the bilinear transform, is employed.

Bilinear Transform Design of IIR Filters

Let H a (s) be the Laplace transform transfer function of an analog filter with frequency response H a (j W) The bilinear transform method obtains the digital filter transfer function H(z) from H a (s) using the substitution

That is,

This maps analog frequency W to digital frequency w according to

thereby warping the frequency response H a (j W) and forcing it to lie between 0 and p for H(e jw) Therefore, toobtain a digital filter with a cutoff frequency of wc it is necessary to design an analog filter with cutoff frequency

This process is referred to as prewarping the analog filter frequency response to compensate for the warping of

the bilinear transform Applying the bilinear transform substitution to this analog filter will then give a digitalfilter that has the desired cutoff frequency

Analog filters and hence IIR digital filters are typically specified in a slightly different fashion than FIR filters.Figure 14.11 illustrates how analog and IIR digital filters are usually specified Notice by comparison to Fig 14.10that the passband ripple in this case never goes above unity, whereas in the FIR case the passband ripple isspecified about unity

= +

-2 1 1