In order to compute the Fourier spectrum of a signal by means of a digital computer, the time- domain signal must be represented by sample values and the spectrum must be computed at a discrete number of frequencies. It can be shown that the following sum gives an approximation to the Fourier spectrum of a signal at frequencies𝑘∕(𝑁𝑇𝑠), 𝑘 = 0, 1, … , 𝑁 − 1:
𝑋𝑘=
𝑁−1∑
𝑛=0
𝑥𝑛𝑒−𝑗2𝜋𝑛𝑘∕𝑁, 𝑘 = 0, 1, ..., 𝑁 − 1 (2.322) where𝑥0, 𝑥1, 𝑥2, ..., 𝑥𝑁−1are𝑁sample values of the signal taken at𝑇𝑠-second intervals for which the Fourier spectrum is desired. The sum(2.322)is called thediscrete Fourier transform (DFT) of the sequence{𝑥𝑛}. According to the sampling theorem, if the samples are spaced by 𝑇𝑠seconds, the spectrum repeats every𝑓𝑠= 𝑇𝑠−1Hz. Since there are𝑁frequency samples in this interval, it follows that the frequency resolution of(2.322)is𝑓𝑠∕𝑁 = 1∕(𝑁𝑇𝑠)≜1∕𝑇. To obtain the sample sequence{𝑥𝑛}from the DFT sequence{𝑋𝑘}, the sum
𝑥𝑛= 1 𝑁
𝑁−1∑
𝑘=0
𝑋𝑘𝑒𝑗2𝜋𝑛𝑘∕𝑁, 𝑘 = 0, 1, 2, … , 𝑁 − 1 (2.323)
is used. That(2.322)and(2.323)form a transform pair can be shown by substituting(2.322) into(2.323)and using the sum formula for a geometric series:
𝑆𝑁≡
𝑁−1∑
𝑘=0
𝑥𝑘=
{1−𝑥𝑁
1−𝑥 , 𝑥≠1
𝑁, 𝑥 = 1 (2.324)
As indicated above, the DFT and inverse DFT are approximations to the true Fourier spectrum of a signal𝑥(𝑡)at the discrete set of frequencies{0, 1∕𝑇 , 2∕𝑇 , … , (𝑁 − 1)∕𝑇 }. The error can be small if the DFT and its inverse are applied properly to a signal. To indicate the approximations involved, we must visualize the spectrum of a sampled signal that is truncated to a finite number of sample values and whose spectrum is then sampled at a discrete number 𝑁 of points. To see the approximations involved, we use the following Fourier-transform theorems:
1. The Fourier transform of an ideal sampling waveform (Example 2.14):
𝑦𝑠(𝑡) =
∑∞ 𝑚=−∞
𝛿(𝑡 − 𝑚𝑇𝑠)⟷𝑓𝑠−1
∑∞ 𝑛=−∞
𝛿(𝑓 − 𝑛𝑓𝑠), 𝑓𝑠= 𝑇𝑠−1 2. The Fourier transform of a rectangular window function:
Π(𝑡∕𝑇 )⟷𝑇 sinc(𝑓𝑇 ) 3. The convolution theorem of Fourier transforms:
𝑥1(𝑡) ∗ 𝑥2(𝑡)⟷𝑋1(𝑓)𝑋2(𝑓) 4. The multiplication theorem of Fourier transforms:
𝑥1(𝑡)𝑥2(𝑡)⟷𝑋1(𝑓) ∗ 𝑋2(𝑓)
The approximations involved are illustrated by the following example.
EXAMPLE 2.33
An exponential signal is to be sampled, the samples truncated to a finite number, and the result represented by a finite number of samples of the Fourier spectrum of the sampled truncated signal. The continuous- time signal and its Fourier transform are
𝑥(𝑡) = 𝑒−|𝑡|∕𝜏⟷𝑋(𝑓) = 2𝜏
1 + 2(𝜋𝑓𝜏)2 (2.325)
This signal and its spectrum are shown in Figure 2.31(a). However, we are representing the signal by sample values spaced by𝑇𝑠seconds, which entails multiplying the original signal by the ideal sampling waveform𝑦𝑠(𝑡), given by(2.114). The resulting spectrum of this sampled signal is the convolution of 𝑋(𝑓)with the Fourier transform of𝑦𝑠(𝑡), given by(2.119), which is𝑌𝑠(𝑓) = 𝑓𝑠∑∞
𝑛=−∞𝛿(𝑓 − 𝑛𝑓𝑠).The result of this convolution in the frequency domain is
𝑋𝑠(𝑓) = 𝑓𝑠
∑∞ 𝑛=−∞
2𝜏
1 + [2𝜋𝜏(𝑓 − 𝑓𝑠)]2 (2.326)
The resulting sampled signal and its spectrum are shown in Figure 2.31(b).
In calculating the DFT, only a𝑇-second chunk of𝑥(𝑡)can be used (𝑁samples spaced by𝑇𝑠= 𝑇 ∕𝑁).
This means that the sampled time-domain signal is effectively multiplied by a window functionΠ(𝑡∕𝑇 ).
In the frequency domain, this corresponds to convolution with the Fourier transform of the rectangular window function, which is𝑇 sinc(𝑓𝑇 ). The resulting windowed, sampled signal and its spectrum are sketched in Figure 2.31(c). Finally, the spectrum is available only at𝑁discrete frequencies separated by the reciprocal of the window duration1∕𝑇. This corresponds to convolution in the time domain with
–2 –1 0 1 –1 0 1
–1 0 1
2
–2 –1 0 1 2
x(t)
X(f)
Xs(f) xs(t)
xs(t)∏( )
f t
f
–1 0 1
–1 –1
–2
–3 0 1 2 3 0 1
f
f t
t
–2 –1 0 1 2 t
(a)
(b)
(c)
(d) t
T
Xsw(f)
Xsp(f) xsp(t)
Figure 2.31
Signals and spectra illustrating the computation of the DFT. (a) Signal to be sampled and its spectrum (𝜏 = 1s). (b) Sampled signal and its spectrum(𝑓𝑠= 1Hz). (c) Windowed, sampled signal and its spectrum(𝑇 = 4+s). (d) Sampled signal spectrum and corresponding periodic repetition of the sampled, windowed signal.
a sequence of delta functions. The resulting signal and spectrum are shown in Figure 2.31(d). It can be seen that unless one is careful, there is indeed a considerable likelihood that the DFT spectrum will look nothing like the spectrum of the original continuous-time signal. Means for minimizing these errors are discussed in several references on the subject.19
■ A little thought will indicate that to compute the complete DFT spectrum of a signal, approximately𝑁2complex multiplications are required in addition to a number of complex additions. It is possible to find algorithms that allow the computation of the DFT spectrum of a signal using only approximately𝑁 log2𝑁complex multiplications, which gives significant computational savings for𝑁large. Such algorithms are referred to as fastFourier-transform (FFT) algorithms. Two main types of FFT algorithms are those based on decimation in time (DIT) and those based ondecimation in frequency (DIF).
Fortunately, FFT algorithms are included in most computer mathematics packages such as MATLABTM, so we do not have to go to the trouble of writing our own FFT programs although it is an instructive exercise to do so. The following computer example computes the FFT of a sampled double-sided exponential pulse and compares spectra of the continuous-time and sampled pulses.
COMPUTER EXAMPLE 2.3
The MATLAB program given below computes the fast Fourier transform (FFT) of a double-sided exponentially decaying signal truncated to−15.5≤𝑡≤15.5sampled each𝑇𝑠= 1s. The periodicity property of the FFT means that the resulting FFT coefficients correspond to a waveform that is the periodic extension of this exponential waveform. The frequency extent of the FFT is[0, 𝑓𝑠(1 − 1∕𝑁)]
with the frequencies above𝑓𝑠∕2corresponding to negative frequencies. Results are shown in Fig. 2.32.
% file: c2ce3
% clf tau = 2;
Ts = 1;
fs = 1/Ts;
ts = -15.5:Ts:15.5;
N = length(ts);
fss = 0:fs/N:fs-fs/N;
xss = exp(-abs(ts)/tau);
Xss = fft(xss);
t = -15.5:.01:15.5;
f = 0:.01:fs-fs/N;
X = 2*fs*tau./(1+(2*pi*f*tau).ˆ2);
subplot(2,1,1), stem(ts, xss) hold on
subplot(2,1,1), plot(t, exp(-abs(t)/tau), ’--’), xlabel(’t, s’), yla- bel(’Signal & samples’), ...
legend(’x(nT s)’, ’x(t)’)
subplot(2,1,2), stem(fss, abs(Xss)) hold on
subplot(2,1,2), plot(f, X, ’--’), xlabel(’f, Hz’), ylabel(’FFT and Fourier transform’)
legend(’|X k|’, ’|X(f)|’)
% End of script file
19Ziemer, Tranter, and Fannin (1998), Chapter 10.
1
0 0.8 0.6 0.4 0.2
0 5
–5 –10
–15
–20 10 15 20
t, s
x(nTs) x(t)
Signal & samples
4
0 1 2 3
0.5 0.4 0.3 0.2 0.1
0 0.6 0.7 0.8 0.9 1
f, Hz
Xk X(f)
FFT and Fourier transform
(a)
(b)
Figure 2.32
(a)𝑥 (𝑡) = exp (−|𝑡|∕𝜏)and samples taken each𝑇𝑠= 1s for𝜏 = 2s; (b) Magnitude of the 32-point FFT of the sampled signal compared with the Fourier transform of𝑥 (𝑡). The spectral plots deviate from each other around𝑓𝑠∕2most due to aliasing.
■
Further Reading
Bracewell (1986) is a text concerned exclusively with Fourier theory and applications. Ziemer, Tranter, and Fannin (1998) and Kamen and Heck (2007) are devoted to continuous- and discrete-time signal and system theory and provide background for this chapter. More elementary books are McClellan, Schafer, and Yoder (2003), Mersereau and Jackson (2006), and Wickert (2013).
Summary
1. Two general classes of signals are deterministic and random. The former can be expressed as completely known functions of time, whereas the amplitudes of random sig- nals must be described probabilistically.
2. A periodic signal of period 𝑇0 is one for which 𝑥(𝑡) = 𝑥(𝑡 + 𝑇0), all𝑡.
3. A single-sided spectrum for a rotating phasor̃𝑥(𝑡) = 𝐴𝑒𝑗(2𝜋𝑓0𝑡+𝜃)shows𝐴(amplitude) and𝜃(phase) versus𝑓
(frequency). The real, sinusoidal signal corresponding to this phasor is obtained by taking the real part of ̃𝑥(𝑡).
A double-sided spectrum results if we think of forming 𝑥(𝑡) =12̃𝑥(𝑡) +12̃𝑥∗(𝑡). Graphs of amplitude and phase (two plots) of this rotating phasor sum versus𝑓 are known as two-sided amplitude and phase spectra, respectively. Such spectral plots are referred to as frequency-domain repre- sentations of the signal𝐴 cos(2𝜋𝑓0𝑡 + 𝜃).
4. The unit impulse function,𝛿(𝑡), can be thought of as a zero-width, infinite-height pulse with unity area. The sifting property,∫−∞∞ 𝑥(𝜆)𝛿(𝜆 − 𝑡0) 𝑑𝜆 = 𝑥(𝑡0), where𝑥(𝑡) is continuous at𝑡 = 𝑡0, is a generalization of the defining relation for a unit impulse. The unit step function,𝑢(𝑡), is the integral of a unit impulse.
5. A signal 𝑥(𝑡) for which 𝐸 =∫−∞∞ |𝑥(𝑡)|2𝑑𝑡 is fi- nite is called anenergy signal. If𝑥(𝑡)is such that 𝑃 = lim𝑇→∞ 1
2𝑇 ∫−𝑇𝑇 |𝑥(𝑡)|2𝑑𝑡is finite, the signal is known as a power signal. Example signals may be either or neither.
6. The complex exponential Fourier series is 𝑥(𝑡) =
∑∞
𝑛=−∞𝑋𝑛exp(𝑗2𝜋𝑛𝑓0𝑡)where 𝑓0= 1∕𝑇0 and (𝑡0, 𝑡0+ 𝑇0)is the expansion interval. The expansion coefficients are given by𝑋𝑛= 𝑇1
0∫𝑡0𝑡0+𝑇0𝑥(𝑡) exp(−𝑗2𝜋𝑛𝑓0𝑡)𝑑𝑡. If𝑥(𝑡) is periodic with period𝑇0, the exponential Fourier series represents𝑥(𝑡)exactly for all𝑡, except at points of dis- continuity where the Fourier sum converges to the mean of the right- and left-handed limits of the signal at the disconinuity.
7. For exponential Fourier series of real signals, the Fourier coefficients obey𝑋𝑛= 𝑋∗−𝑛, which implies that
|𝑋𝑛|=|𝑋−𝑛|and∕𝑋𝑛= −∕𝑋−𝑛. Plots of|𝑋𝑛|and ∕𝑋𝑛 versus 𝑛𝑓0 are referred to as the discrete, double-sided amplitude and phase spectra, respectively, of𝑥(𝑡). If𝑥(𝑡) is real, the amplitude spectrum is even and the phase spec- trum is odd as functions of𝑛𝑓0.
8. Parseval’s theorem for periodic signals is 1
𝑇0∫𝑇0|𝑥 (𝑡)|2 𝑑𝑡 =
∑∞ 𝑛=−∞||𝑋𝑛||2 9. The Fourier transform of a signal𝑥(𝑡)is
𝑋(𝑓) =∫
∞
−∞𝑥(𝑡)𝑒−𝑗2𝜋𝑓𝑡𝑑𝑡 and the inverse Fourier transform is
𝑥(𝑡) =∫
∞
−∞𝑋(𝑓)𝑒𝑗2𝜋𝑓𝑡𝑑𝑓
For real signals, |𝑋(𝑓)|=|𝑋(−𝑓)| and ∕𝑋(𝑓) =
−∕𝑋(−𝑓).
10. Plots of|𝑋(𝑓)|and∕𝑋(𝑓)versus𝑓 are referred to as the double-sided amplitude and phase spectra, respec- tively, of𝑥(𝑡). As functions of frequency, the amplitude spectrum of a real signal is even and its phase spectrum is odd.
11. The energy of a signal is
∫
∞
−∞|𝑥 (𝑡)|2 𝑑𝑡 =∫
∞
−∞|𝑋 (𝑓)|2𝑑𝑓
This is known asRayleigh’s energy theorem. The energy spectral density of a signal is𝐺(𝑓) =|𝑋(𝑓)|2. It is the density of energy with frequency of the signal.
12. The convolution of two signals,𝑥1(𝑡)and𝑥2(𝑡), is 𝑥(𝑡) = 𝑥1∗ 𝑥2=∫
∞
−∞𝑥1(𝜆)𝑥2(𝑡 − 𝜆) 𝑑𝜆
=∫
∞
−∞𝑥1(𝑡 − 𝜆)𝑥2(𝜆) 𝑑𝜆
The convolution theorem of Fourier transforms states that 𝑋(𝑓) = 𝑋1(𝑓)𝑋2(𝑓), where𝑋(𝑓),𝑋1(𝑓), and𝑋2(𝑓)are the Fourier transforms of𝑥(𝑡),𝑥1(𝑡), and𝑥2(𝑡), respec- tively.
13. The Fourier transform of a periodic signal can be obtained formally by Fourier-transforming its exponential Fourier series term by term using𝐴𝑒𝑗2𝜋𝑓0𝑡⟷𝐴𝛿(𝑓 − 𝑓0), even though, mathematically speaking, Fourier transforms of power signals do not exist. A more convenient approach is to convolve a pulse-type signal,𝑝 (𝑡), with the ideal sam- pling waveform to get a periodic signal of the form𝑥 (𝑡) = 𝑝 (𝑡) ∗∑∞
𝑚=−∞𝛿( 𝑡 − 𝑚𝑇𝑠)
; it follows that its Fourier trans- form is 𝑋 (𝑓) =∑∞
𝑛=−∞𝑓𝑠𝑃( 𝑛𝑓𝑠)
𝛿( 𝑓 − 𝑛𝑓𝑠)
where 𝑃 (𝑓)is the Fourier transform of𝑝 (𝑡)and𝑓𝑠= 1∕𝑇𝑠. It follows that the Fourier coeffcients are𝑋𝑛= 𝑓𝑠𝑃(
𝑛𝑓𝑠) . 14. The power spectrum 𝑆(𝑓)of a power signal𝑥(𝑡) is a real, even, nonnegative function that integrates to give total average power:⟨
𝑥2(𝑡)⟩
=∫−∞∞𝑆(𝑓) 𝑑𝑓 where
⟨𝑤 (𝑡)⟩≜lim𝑇→∞ 1
2𝑇 ∫−𝑇𝑇 𝑤 (𝑡) 𝑑𝑡.The time-average auto- correlation function of a power signal is defined as𝑅 (𝜏) =
⟨𝑥(𝑡)𝑥(𝑡 + 𝜏)⟩. The Wiener--Khinchine theorem states that 𝑆(𝑓)and𝑅(𝜏)are Fourier-transform pairs.
15. A linear system, denoted operationally as(⋅), is one for which superposition holds; that is, if𝑦1=(𝑥1)and 𝑦2=(𝑥2), then (𝛼1𝑥1+ 𝛼2𝑥2) = 𝛼1𝑦1+ 𝛼2𝑦2, where 𝑥1and𝑥2are inputs,𝑦1and𝑦2are outputs (the time vari- able 𝑡 is suppressed for simplicity), and 𝛼1 and 𝛼2 are arbitrary constants. A system is fixed, or time-invariant, if, given𝑦(𝑡) =[𝑥(𝑡)], the input𝑥(𝑡 − 𝑡0)results in the output𝑦(𝑡 − 𝑡0).
16. The impulse responseℎ(𝑡)of a linear time-invariant (LTI) system is its response to an impulse applied at𝑡 = 0: ℎ(𝑡) =[𝛿(𝑡)]. The output of an LTI system to an input 𝑥(𝑡)is given by𝑦(𝑡) = ℎ(𝑡) ∗ 𝑥(𝑡) =∫−∞∞ ℎ (𝜏) 𝑥 (𝑡 − 𝜏) 𝑑𝜏.
17. Acausal system is one that does not anticipate its input. For such an LTI system,ℎ(𝑡) = 0for𝑡 < 0. Astable system is one for which every bounded input results in a bounded output. An LTI system is stable if and only if
∫−∞∞ |ℎ(𝑡)|𝑑𝑡 < ∞.
18. The frequency response function𝐻(𝑓)of an LTI system is the Fourier transform ofℎ(𝑡). The Fourier trans- form of the system output 𝑦(𝑡) due to an input 𝑥(𝑡) is 𝑌 (𝑓) = 𝐻(𝑓)𝑋(𝑓), where𝑋(𝑓)is the Fourier transform of the input.|𝐻(𝑓)|=|𝐻(−𝑓)|is called theamplitude response of the system and∕𝐻(𝑓) = −∕𝐻(−𝑓)is called thephase response.
19. For a fixed linear system with a periodic input, the Fourier coefficients of the output are given by 𝑌𝑛= 𝐻(𝑛𝑓0)𝑋𝑛, where{𝑋𝑛} represents the Fourier co- efficients of the input.
20. Input and output spectral densities for a fixed linear system are related by
𝐺𝑦(𝑓) =|𝐻(𝑓)|2𝐺𝑥(𝑓) (energy signals) 𝑆𝑦(𝑓) =|𝐻(𝑓)|2𝑆𝑥(𝑓) (power signals) 21. A system is distortionless if its output looks like its input except for a time delay and amplitude scaling:
𝑦(𝑡) = 𝐻0𝑥(𝑡 − 𝑡0). The frequency response function of a distortionless system is 𝐻(𝑓) = 𝐻0𝑒−𝑗2𝜋𝑓𝑡0. Such a sys- tem’s amplitude response is |𝐻(𝑓)|= 𝐻0 and its phase response is ∕𝐻(𝑓) = −2𝜋𝑡0𝑓 over the band of frequen- cies occupied by the input. Three types of distortion that a system may introduce are amplitude, phase (or delay), and nonlinear, depending on whether|𝐻(𝑓)|≠constant,
∕𝐻(𝑓)≠−constant×𝑓, or the system is nonlinear, respec- tively. Two other important properties of a linear system are the group and phase delays. These are defined by
𝑇𝑔(𝑓) = −1 2𝜋
𝑑𝜃(𝑓)
𝑑𝑓 and 𝑇𝑝(𝑓) = −𝜃 (𝑓) 2𝜋𝑓 respectively, in which𝜃(𝑓)is the phase response of the LTI system. Phase distortionless systems have equal group and phase delays (constant).
22. Ideal filters are convenient in communication system analysis, even though they are noncausal. Three types of ideal filters are lowpass, bandpass, and highpass. Through- out their passbands, ideal filters have constant amplitude response and linear phase response. Outside their pass- bands, ideal filters perfectly reject all spectral components of the input.
23. Approximations to ideal filters are Butterworth, Chebyshev, and Bessel filters. The first two are attempts at approximating the amplitude response of an ideal filter, and the latter is an attempt to approximate the linear phase response of an ideal filter.
24. An inequality relating the duration 𝑇 of a pulse and its single-sided bandwidth𝑊 is𝑊 ≥1∕ (2𝑇 ). Pulse risetime 𝑇𝑅 and signal bandwidth are related approxi- mately by𝑊 = 1∕(
2𝑇𝑅)
. These relationships hold for the lowpass case. For bandpass filters and signals, the re- quired bandwidth is doubled, and the risetime is that of the envelope of the signal.
25. The sampling theorem for lowpass signals of band- width𝑊 states that a signal can be perfectly recovered by lowpass filtering from sample values taken at a rate of 𝑓𝑠> 2𝑊samples per second. The spectrum of an impulse- sampled signal is
𝑋𝛿(𝑓) = 𝑓𝑠
∑∞ 𝑛=−∞
𝑋(𝑓 − 𝑛𝑓𝑠)
where𝑋(𝑓) is the spectrum of the original signal. For bandpass signals, lower sampling rates than specified by the lowpass sampling theorem may be possible.
26. The Hilbert transform ̂𝑥(𝑡)of a signal 𝑥(𝑡)corre- sponds to a−90◦phase shift of all the signal’s positive- frequency components. Mathematically,
̂𝑥 (𝑡) =∫
∞
−∞
𝑥 (𝜆) 𝜋 (𝑡 − 𝜆)𝑑𝜆
In the frequency domain,𝑋(𝑓) = −𝑗̂ sgn(𝑓)𝑋(𝑓), where sgn(𝑓) is the signum function, 𝑋(𝑓) =ℑ[𝑥(𝑡)], and 𝑋(𝑓) =̂ ℑ[ ̂𝑥(𝑡)]. The Hilbert transform of cos 𝜔0𝑡 is sin 𝜔0𝑡, and the Hilbert transform ofsin 𝜔0𝑡is− cos 𝜔0𝑡. The power (or energy) in a signal and its Hilbert transform are equal. A signal and its Hilbert transform are orthogonal in the range(−∞, ∞). If𝑚(𝑡)is a lowpass signal and𝑐(𝑡) is a highpass signal with nonoverlapping spectra,
𝑚(̂𝑡)𝑐(𝑡) = 𝑚(𝑡)̂𝑐(𝑡)
The Hilbert transform can be used to define the analytic signal
𝑧(𝑡) = 𝑥(𝑡) ± 𝑗̂𝑥(𝑡)
The magnitude of the analytic signal,|𝑧(𝑡)|, is the envelope of the real signal𝑥(𝑡). The Fourier transform of an ana- lytic signal,𝑍(𝑓), is identically zero for𝑓 < 0or𝑓 > 0, respectively, depending on whether the+sign or−sign is chosen for the imaginary part of𝑧(𝑡).
27. The complex envelope ̃𝑥(𝑡)of a bandpass signal is defined by
𝑥(𝑡) + 𝑗̂𝑥(𝑡) = ̃𝑥(𝑡)𝑒𝑗2𝜋𝑓0𝑡
where𝑓0is the reference frequency for the signal. Simi- larly, the complex envelopẽℎ(𝑡)of the impulse response of a bandpass system is defined by
ℎ(𝑡) + 𝑗̂ℎ(𝑡) = ̃ℎ(𝑡)𝑒𝑗2𝜋𝑓0𝑡
The complex envelope of the bandpass system output is conveniently obtained in terms of the complex envelope of the output, which can be found from either of the oper- ations
̃𝑦(𝑡) = ̃ℎ(𝑡) ∗ ̃𝑥(𝑡) or
̃𝑦(𝑡) =ℑ−1[
𝐻(𝑓) ̃̃ 𝑋(𝑓)]
where𝐻(𝑓)̃ and𝑋(𝑓)̃ are the Fourier transforms of̃ℎ(𝑡) and̃𝑥(𝑡), respectively. The actual (real) output is then given by
𝑦(𝑡) =1 2Re [
̃𝑦(𝑡)𝑒𝑗2𝜋𝑓0𝑡]
28. The discrete Fourier transform (DFT) of a signal se- quence{
𝑥𝑛}
is defined as 𝑋𝑘=
𝑁−1∑
𝑛=0
𝑥𝑛𝑒𝑗2𝜋𝑛𝑘∕𝑁= DFT [ {𝑥𝑛}]
, 𝑘 = 0, 1, ..., 𝑁 − 1 and the inverse DFT can be found from
𝑥𝑛= 1 𝑁 DFT [
{𝑋𝑘∗}]∗
, 𝑘 = 0, 1, ..., 𝑁 − 1 The DFT can be used to digitally compute spectra of sam- pled signals and to approximate operations carried out by the normal Fourier transform, for example, filtering.
Drill Problems
2.1 Find the fundamental periods of the following sig- nals:
(a) 𝑥1(𝑡) = 10 cos (5𝜋𝑡)
(b) 𝑥2(𝑡) = 10 cos (5𝜋𝑡) + 2 sin (7𝜋𝑡)
(c) 𝑥3(𝑡) = 10 cos (5𝜋𝑡) + 2 sin (7𝜋𝑡) + 3 cos (6.5𝜋𝑡) (d) 𝑥4(𝑡) = exp (𝑗6𝜋𝑡)
(e) 𝑥5(𝑡) = exp (𝑗6𝜋𝑡) + exp(−𝑗6𝜋𝑡) (f) 𝑥6(𝑡) = exp (𝑗6𝜋𝑡) + exp (𝑗7𝜋𝑡)
2.2 Plot the double-sided amplitude and phase spectra of the periodic signals given in Drill Problem 2.1.
2.3 Plot the single-sided amplitude and phase spectra of the periodic signals given in Drill Problem 2.1.
2.4 Evaluate the following integrals:
(a) 𝐼1=∫−1010𝑢 (𝑡) 𝑑𝑡 (b) 𝐼2=∫−1010𝛿 (𝑡 − 1) 𝑢 (𝑡) 𝑑𝑡 (c) 𝐼3=∫−1010𝛿 (𝑡 + 1) 𝑢 (𝑡) 𝑑𝑡 (d) 𝐼4=∫−1010𝛿 (𝑡 − 1) 𝑡2𝑑𝑡 (e) 𝐼5=∫−1010𝛿 (𝑡 + 1) 𝑡2𝑑𝑡 (f) 𝐼6=∫−1010𝑡2𝑢 (𝑡 − 1) 𝑑𝑡
2.5 Find the powers and energies of the following sig- nals (0 and∞are possible answers):
(a) 𝑥1(𝑡) = 2𝑢 (𝑡) (b) 𝑥2(𝑡) = 3Π(
𝑡−1 2
)
(c) 𝑥3(𝑡) = 2Π(
𝑡−3 4
)
(d) 𝑥4(𝑡) = cos (2𝜋𝑡) (e) 𝑥5(𝑡) = cos (2𝜋𝑡) 𝑢 (𝑡) (f) 𝑥6(𝑡) = cos2(2𝜋𝑡) + sin2(2𝜋𝑡)
2.6 Tell whether or not the following can be Fourier coefficients of real signals (give reasons for your answers):
(a) 𝑋1= 1 + 𝑗; 𝑋−1= 1 − 𝑗;all other Fourier coef- ficients are 0
(b) 𝑋1= 1 + 𝑗; 𝑋−1= 2 − 𝑗;all other Fourier coef- ficients are 0
(c) 𝑋1= exp (−𝑗𝜋∕2) ; 𝑋−1= exp (𝑗𝜋∕2) ;all other Fourier coefficients are 0
(d) 𝑋1= exp (𝑗3𝜋∕2) ; 𝑋−1= exp (𝑗𝜋∕2) ;all other Fourier coefficients are 0
(e) 𝑋1= exp (𝑗3𝜋∕2) ; 𝑋−1= exp(𝑗5𝜋∕2);all other Fourier coefficients are 0
2.7 By invoking uniqueness of the Fourier series, give the complex exponential Fourier series coefficients for the following signals:
(a) 𝑥1(𝑡) = 1 + cos (2𝜋𝑡) (b) 𝑥2(𝑡) = 2 sin (2𝜋𝑡)
(c) 𝑥3(𝑡) = 2 cos (2𝜋𝑡) + 2 sin (2𝜋𝑡) (d) 𝑥4(𝑡) = 2 cos (2𝜋𝑡) + 2 sin (4𝜋𝑡)
(e) 𝑥5(𝑡) = 2 cos (2𝜋𝑡) + 2 sin (4𝜋𝑡) + 3 cos (6𝜋𝑡) 2.8 Tell whether the following statements are true or false and why:
(a) A triangular wave has only odd harmonics in its Fourier series.
(b) The spectral content of a pulse train has more higher-frequency content the longer the pulse width.
(c) A full rectified sine wave has a fundamental fre- quency, which is half that of the original sinusoid that was rectified.
(d) The harmonics of a square wave decrease faster with the harmonic number𝑛than those of a tri- angular wave.
(e) The delay of a pulse train affects its amplitude spectrum.
(f) The amplitude spectra of a half-rectified sine wave and a half-rectified cosine wave are identi- cal.
2.9 Given the Fourier-transform pairs Π (𝑡)⟷ sinc(𝑓)andΛ (𝑡)⟷sinc2(𝑓), use appropriate Fourier- transform theorems to find Fourier transforms of the fol- lowing signals. Tell which theorem(s) you used in each case. Sketch signals and transforms.
(a) 𝑥1(𝑡) = Π (2𝑡) (b) 𝑥2(𝑡) =sinc2(4𝑡) (c) 𝑥3(𝑡) = Π (2𝑡) cos (6𝜋𝑡) (d) 𝑥4(𝑡) = Λ(
𝑡−3 2
)
(e) 𝑥5(𝑡) = Π (2𝑡) ⋆ Π (2𝑡) (f) 𝑥6(𝑡) = Π (2𝑡) exp (𝑗4𝜋𝑡) (g) 𝑥7(𝑡) = Π(
𝑡 2
)+ Λ (𝑡)
(h) 𝑥8(𝑡) =𝑑Λ(𝑡)𝑑𝑡 (i) 𝑥9(𝑡) = Π(
𝑡 2
)Λ (𝑡)
2.10 Obtain the Fourier transform of the signal𝑥 (𝑡) =
∑∞
𝑚=−∞Λ (𝑡 − 3𝑚). Sketch the signal and its transform.
2.11 Obtain the power spectral densities corresponding to the autocorrelation functions given below. Verify in each case that the power spectral density integrates to the
total average power [i.e.,𝑅 (0)]. Provide a sketch of each autocorrelation function and corresponding power spectral density.
(a) 𝑅1(𝜏) = 3Λ (𝜏∕2) (b) 𝑅2(𝜏) = 2 cos (4𝜋𝜏)
(c) 𝑅3(𝜏) = 2Λ (𝜏∕2) cos (4𝜋𝜏) (d) 𝑅4(𝜏) = exp (−2|𝜏|)
(e) 𝑅5(𝜏) = 1 + cos (2𝜋𝜏)
2.12 Obtain the impulse response of a system with frequency response function 𝐻 (𝑓) = 2∕ (3 + 𝑗2𝜋𝑓) + 1∕ (2 + 𝑗2𝜋𝑓). Plot the impulse response and the ampli- tude and phase responses.
2.13 Tell whether or not the following systems are (1) stable and (2) causal. Give reasons for your answers.
(a) ℎ1(𝑡) = 3∕ (4 +|𝑡|) (b) 𝐻2(𝑓) = 1 + 𝑗2𝜋𝑓
(c) 𝐻3(𝑓) = 1∕ (1 + 𝑗2𝜋𝑓) (d) ℎ4(𝑡) = exp (−2|𝑡|)
(e) ℎ5(𝑡) =[
2 exp (−3𝑡) + exp (−2𝑡)] 𝑢 (𝑡)
2.14 Find the phase and group delays for the following systems.
(a) ℎ1(𝑡) = exp (−2𝑡) 𝑢 (𝑡) (b) 𝐻2(𝑓) = 1 + 𝑗2𝜋𝑓
(c) 𝐻3(𝑓) = 1∕ (1 + 𝑗2𝜋𝑓) (d) ℎ4(𝑡) = 2𝑡 exp (−3𝑡) 𝑢 (𝑡)
2.15 A filter has frequency response function 𝐻 (𝑓) =
[ Π
(𝑓 30
)
+ Π (𝑓
10 )]
exp[
−𝑗𝜋𝑓Π (𝑓∕15) ∕20] The input is𝑥 (𝑡) = 2 cos(
2𝜋𝑓1𝑡) + cos(
2𝜋𝑓2𝑡) . For the values of𝑓1and𝑓2 given below tell whether there is (1) no distortion, (2) amplitude distortion, (3) phase or delay distortion, or (4) both amplitude and phase (delay) distor- tion.
(a) 𝑓1= 2Hz and𝑓2= 4Hz (b) 𝑓1= 2Hz and𝑓2= 6Hz (c) 𝑓1= 2Hz and𝑓2= 8Hz (d) 𝑓1= 6Hz and𝑓2= 7Hz (e) 𝑓1= 6Hz and𝑓2= 8Hz (f) 𝑓1= 8Hz and𝑓2= 16Hz
2.16 A filter has input-output transfer characteristic given by 𝑦 (𝑡) = 𝑥 (𝑡) + 𝑥2(𝑡). With the input 𝑥 (𝑡) = cos(
2𝜋𝑓1𝑡) + cos(
2𝜋𝑓2𝑡)
tell what frequency compo- nents will appear at the output. Which are distortion terms?
2.17 A filter has frequency response function
𝐻 (𝑗2𝜋𝑓) = 2
− (2𝜋𝑓)2+ 𝑗4𝜋𝑓 + 1. Find its 10% to 90%
risetime.
2.18 The signal𝑥 (𝑡) = cos( 2𝜋𝑓1𝑡)
is sampled at𝑓𝑠= 9 samples per second. Give the lowest frequency present in the sampled signal spectrum for the following values of 𝑓1:
(a) 𝑓1= 2Hz (b) 𝑓1= 4Hz (c) 𝑓1= 6Hz (d) 𝑓1= 8Hz
(e) 𝑓1= 10Hz (f) 𝑓1= 12Hz
2.19 Give the Hilbert transforms of the following signals:
(a) 𝑥1(𝑡) = cos (4𝜋𝑡) (b) 𝑥2(𝑡) = sin (6𝜋𝑡)
(c) 𝑥3(𝑡) = exp (𝑗5𝜋𝑡) (d) 𝑥4(𝑡) = exp (−𝑗8𝜋𝑡)
(e) 𝑥5(𝑡) = 2 cos2(4𝜋𝑡) (f) 𝑥6(𝑡) = cos (2𝜋𝑡) cos (10𝜋𝑡) (g) 𝑥7(𝑡) = 2 sin (4𝜋𝑡) cos (4𝜋𝑡)
2.20 Obtain the analytic signal and complex envelope of the signal𝑥 (𝑡) = cos (10𝜋𝑡), where𝑓0= 6Hz.
Problems
Section 2.1
2.1 Sketch the single-sided and double-sided amplitude and phase spectra of the following signals:
(a) 𝑥1(𝑡) = 10 cos(4𝜋𝑡 + 𝜋∕8) + 6 sin(8𝜋𝑡 + 3𝜋∕4) (b) 𝑥2(𝑡) = 8 cos(2𝜋𝑡 + 𝜋∕3) + 4 cos(6𝜋𝑡 + 𝜋∕4) (c) 𝑥3(𝑡) = 2 sin(4𝜋𝑡 + 𝜋∕8) + 12 sin(10𝜋𝑡) (d) 𝑥4(𝑡) = 2 cos(7𝜋𝑡 + 𝜋∕4) + 3 sin(18𝜋𝑡 + 𝜋∕2) (e) 𝑥5(𝑡) = 5 sin(2𝜋𝑡) + 4 cos(5𝜋𝑡 + 𝜋∕4)
(f) 𝑥6(𝑡) = 3 cos(4𝜋𝑡 + 𝜋∕8) + 4 sin(10𝜋𝑡 + 𝜋∕6) 2.2 A signal has the double-sided amplitude and phase spectra shown in Figure 2.33. Write a time-domain expres- sion for the signal.
Amplitude 4 2
2 0 –2
–4 4 f
Phase
–2 2 4
–4 f
4 –π
2 π
2 –π 4 π
Figure 2.33
2.3 The sum of two or more sinusoids may or may not be periodic depending on the relationship of their separate frequencies. For the sum of two sinusoids, let the frequen- cies of the individual terms be𝑓1 and𝑓2, respectively.
For the sum to be periodic,𝑓1 and𝑓2 must be commen- surable; i.e., there must be a number𝑓0contained in each an integral number of times. Thus, if𝑓0is the largest such number,
𝑓1= 𝑛1𝑓0and𝑓2= 𝑛2𝑓0
where𝑛1 and𝑛2 are integers;𝑓0 is the fundamental fre- quency. Which of the signals given below are periodic?
Find the periods of those that are periodic.
(a) 𝑥1(𝑡) = 2 cos(2𝑡) + 4 sin(6𝜋𝑡) (b) 𝑥2(𝑡) = cos(6𝜋𝑡) + 7 cos(30𝜋𝑡)
(c) 𝑥3(𝑡) = cos(4𝜋𝑡) + 9 sin(21𝜋𝑡)
(d) 𝑥4(𝑡) = 3 sin(4𝜋𝑡) + 5 cos(7𝜋𝑡) + 6 sin(11𝜋𝑡) (e) 𝑥5(𝑡) = cos(17𝜋𝑡) + 5 cos(18𝜋𝑡)
(f) 𝑥6(𝑡) = cos(2𝜋𝑡) + 7 sin(3𝜋𝑡) (g) 𝑥7(𝑡) = 4 cos(7𝜋𝑡) + 5 cos(11𝜋𝑡) (h) 𝑥8(𝑡) = cos(120𝜋𝑡) + 3 cos(377𝑡)
(i) 𝑥9(𝑡) = cos(19𝜋𝑡) + 2 sin(21𝜋𝑡) (j) 𝑥10(𝑡) = 5 cos(6𝜋𝑡) + 6 sin(7𝜋𝑡)
2.4 Sketch the single-sided and double-sided amplitude and phase spectra of
(a) 𝑥1(𝑡) = 5 cos(12𝜋𝑡 − 𝜋∕6) (b) 𝑥2(𝑡) = 3 sin(12𝜋𝑡) + 4 cos(16𝜋𝑡) (c) 𝑥3(𝑡) = 4 cos (8𝜋𝑡) cos (12𝜋𝑡)
(Hint: Use an appropriate trigonometric identity.) (d) 𝑥4(𝑡) = 8 sin (2𝜋𝑡) cos2(5𝜋𝑡)
(Hint: Use appropriate trigonometric identities.) (e) 𝑥5(𝑡) = cos(6𝜋𝑡) + 7 cos(30𝜋𝑡)
(f) 𝑥6(𝑡) = cos(4𝜋𝑡) + 9 sin(21𝜋𝑡)
(g) 𝑥7(𝑡) = 2 cos(4𝜋𝑡) + cos(6𝜋𝑡) + 6 sin(17𝜋𝑡) 2.5
(a) Show that the function 𝛿𝜖(𝑡) sketched in Fig- ure 2.4(b) has unity area.
(b) Show that
𝛿𝜖(𝑡) = 𝜖−1𝑒−𝑡∕𝜖𝑢(𝑡)
has unity area. Sketch this function for𝜖 = 1,12, and 14. Comment on its suitability as an approxi- mation for the unit impulse function.
(c) Show that a suitable approximation for the unit impulse function as𝜖→0is given by
𝛿𝜖(𝑡) =
{𝜖−1(1 −|𝑡|∕𝜖) , |𝑡|≤𝜖
0, otherwise
2.6 Use the properties of the unit impulse function given after(2.14)to evaluate the following relations.
(a) ∫−∞∞[𝑡2+ exp(−2𝑡)]𝛿(2𝑡 − 5) 𝑑𝑡 (b) ∫−1010+−(𝑡2+ 1)[∑∞
𝑛=−∞𝛿 (𝑡 − 5𝑛)]
𝑑𝑡 (Note: 10+ means just to the right of10;−10− means just
to the left of−10) (c) 10𝛿 (𝑡) + 𝐴𝑑𝛿(𝑡)𝑑𝑡 + 3𝑑2𝛿(𝑡)
𝑑𝑡2 = 𝐵𝛿 (𝑡) + 5𝑑𝛿(𝑡)𝑑𝑡 + 𝐶𝑑2𝑑𝑡𝛿(𝑡)2 ; find𝐴,𝐵, and𝐶
(d) ∫−211[𝑒−4𝜋𝑡+ tan(10𝜋𝑡)]𝛿(4𝑡 + 3) 𝑑𝑡 (e) ∫−∞∞[cos(5𝜋𝑡) + 𝑒−3𝑡]𝑑𝛿2(𝑡−2)
𝑑𝑡2 𝑑𝑡
2.7 Which of the following signals are periodic and which are aperiodic? Find the periods of those that are periodic. Sketch all signals.
(a) 𝑥𝑎(𝑡) = cos(5𝜋𝑡) + sin(7𝜋𝑡) (b) 𝑥𝑏(𝑡) =∑∞
𝑛=0Λ(𝑡 − 2𝑛) (c) 𝑥𝑐(𝑡) =∑∞
𝑛=−∞Λ(𝑡 − 2𝑛) (d) 𝑥𝑑(𝑡) = sin(3𝑡) + cos(2𝜋𝑡)
(e) 𝑥𝑒(𝑡) =∑∞
𝑛=−∞Π(𝑡 − 3𝑛) (f) 𝑥𝑓(𝑡) =∑∞
𝑛=0Π(𝑡 − 3𝑛)
2.8 Write the signal𝑥(𝑡) = cos(6𝜋𝑡) + 2 sin(10𝜋𝑡)as (a) The real part of a sum of rotating phasors.
(b) A sum of rotating phasors plus their complex conjugates.
(c) From your results in parts (a) and (b), sketch the single-sided and double-sided amplitude and phase spectra of𝑥(𝑡).
Section 2.2
2.9 Find the normalized power for each signal below that is a power signal and the normalized energy for each signal that is an energy signal. If a signal is neither a power signal nor an energy signal, so designate it. Sketch each signal (𝛼is a positive constant).
(a) 𝑥1(𝑡) = 2 cos(4𝜋𝑡 + 2𝜋∕3) (b) 𝑥2(𝑡) = 𝑒−𝛼𝑡𝑢(𝑡)
(c) 𝑥3(𝑡) = 𝑒𝛼𝑡𝑢(−𝑡) (d) 𝑥4(𝑡) =(
𝛼2+ 𝑡2)−1∕2
(e) 𝑥5(𝑡) = 𝑒−𝛼|𝑡|
(f) 𝑥6(𝑡) = 𝑒−𝛼𝑡𝑢(𝑡) − 𝑒−𝛼(𝑡−1)𝑢(𝑡 − 1)
2.10 Classify each of the following signals as an energy signal or as a power signal by calculating𝐸, the energy, or𝑃, the power(𝐴, 𝐵, 𝜃, 𝜔,and𝜏are positive constants).
(a) 𝑥1(𝑡) = 𝐴|sin (𝜔𝑡 + 𝜃)| (b) 𝑥2(𝑡) = 𝐴𝜏∕√
𝜏 + 𝑗𝑡, 𝑗 =√
−1 (c) 𝑥3(𝑡) = 𝐴𝑡𝑒−𝑡∕𝜏𝑢 (𝑡)
(d) 𝑥4(𝑡) = Π(𝑡∕𝜏) + Π(𝑡∕2𝜏)