CHAPTER 5 Frequency Analysis: The Fourier Transform
5.9 What Have We Accomplished? What Is Next?
You should by now have a very good understanding of the frequency representation of signals and systems. In this chapter, we have unified the treatment of periodic and nonperiodic signals and their spectra, and consolidated the concept of frequency response of a linear time-invariant system.
Basic properties of the Fourier transform and important Fourier pairs are given in Tables 5.1 and 5.2. Two significant applications are in filtering and communications. We introduced the basics of filtering in this chapter and will expand on them in Chapter 6. The fundamentals of modulation provided in this chapter will be illustrated in Chapter 6 where we will consider their application in communications.
Certainly the next step is to find out where the Laplace and the Fourier analyses apply, which will be done in Chapter 6. After that, we will go into discrete-time signals and systems. We will show that sampling, quantization, and coding bridge the continuous-time and the digital signal processing, and that transformations similar to the Laplace and the Fourier transforms will permit us to do processing of discrete–time signals and systems.
PROBLEMS
5.1. Fourier series versus Fourier transform—MATLAB
The connection between the Fourier series and the Fourier transform can be seen by considering what happens when the period of a periodic signal increases to a point at which the periodicity is not clear as only one period is seen. Consider a train of pulsesx(t)with a period T0=2, and a period ofx(t)is x1(t)=u(t+0.5)−u(t−0.5). LetT0be increased to4, 8, and16.
Problems 351
Table 5.1 Basic Properties of the Fourier Transform
Time Domain Frequency Domain
Signals and constants x(t),y(t),z(t),α,β X(),Y(),Z() Linearity αx(t)+βy(t) αX()+βY() Expansion/contraction in time x(αt),α6=0 |α|1 X α
Reflection x(−t) X(−)
Parseval’s energy relation Ex=R∞
−∞|x(t)|2dt Ex= 21πR∞
−∞|X()|2d
Duality X(t) 2πx(−)
Time differentiation dndtx(nt), n≥1, integer (j)nX() Frequency differentiation −jtx(t) dX()d
Integration Rt
−∞x(t0)dt0 Xj()+πX(0)δ()
Time shifting x(t−α) e−jαX()
Frequency shifting ej0tx(t) X(−0)
Modulation x(t)cos(ct) 0.5[X(−c)+X(+c)] Periodic signals x(t)=P
kXkejk0t X()=P
k2πXkδ(−k0)
Symmetry x(t)real |X()| = |X(−)|
∠X()= −∠X(−) Convolution in time z(t)=[x∗y](t) Z()=X()Y() Windowing/multiplication x(t)y(t) 21π[X∗Y]()
Cosine transform x(t)even X()=R∞
−∞x(t)cos(t)dt,real
Sine transform x(t)odd X()= −jR∞
−∞x(t)sin(t)dt, imaginary
Table 5.2 Fourier Transform Pairs
Function of Time Function of
1 δ(t) 1
2 δ(t−τ) e−jτ
3 u(t) j1+πδ()
4 u(−t) −j1+πδ()
5 sgn(t)=2[u(t)−0.5] j2 6 A, −∞<t<∞ 2πAδ() 7 Ae−atu(t), a>0 j+aA 8 Ate−atu(t), a>0 (j+Aa)2 9 e−a|t|, a>0 a22a+2
10 cos(0t), −∞<t<∞ π[δ(−0)+δ(+0)] 11 sin(0t), −∞<t<∞ −jπ[δ(−0)−δ(+0)] 12 A[u(t+τ)−u(t−τ)],τ >0 2Aτsinτ(τ)
13 sin(πt0t) u(+0)−u(−0) 14 x(t)cos(0t) 0.5[X(−0)+X(+0)]
(a) Find the Fourier series coefficientX0for each of the values ofT0and indicate how it changes for the different values ofT0.
(b) Find the Fourier series coefficients forx(t)and carefully plot the magnitude line spectrum for each of the values ofT0. Explain what is happening in these spectra.
(c) If you were to letT0 be very large, what would you expect to happen to the Fourier coefficients?
Explain.
(d) Write a MATLAB script that simulates the conversion from the Fourier series to the Fourier transform of a sequence of rectangular pulses as the period is increased. The Fourier coefficients need to be multiplied by the period so that they do not become insignificant. Plot usingstemthe magnitude line spectrum for pulse sequences with periodsT0from 4 to 62.
5.2. Fourier transform from Laplace transform—MATLAB
The Fourier transform of finite-support signals, which are absolutely integrable or finite energy, can be obtained from their Laplace transform rather than doing the integral. Consider the following signals:
x1(t)=u(t+0.5)−u(t−0.5) x2(t)=sin(2πt)[u(t)−u(t−0.5)]
x3(t)=r(t+1)−2r(t)+r(t−1)
(a) Plot each of the signals.
(b) Find the Fourier transforms{Xi()}fori=1, 2, and3using the Laplace transform.
(c) Use MATLAB’s symbolic functionfourierto compute the Fourier transform of the given signals. Plot the magnitude spectrum corresponding to each of the signals.
5.3. Fourier transform from Laplace transform of infinite-support signals—MATLAB
For signals with infinite support, their Fourier transforms cannot be derived from the Laplace transform unless they are absolutely integrable or the region of convergence of the Laplace transform contains thej
axis. Consider the signalx(t)=2e−2|t|. (a) Plot the signalx(t)for−∞<t<∞.
(b) Use the evenness of the signal to find the integral Z∞
−∞
|x(t)|dt
and determine whether this signal is absolutely integrable or not.
(c) Use the integral definition of the Fourier transform to findX().
(d) Use the Laplace transform ofx(t)to verify the above found Fourier transform.
(e) Use MATLAB’s symbolic functionfourierto compute the Fourier transform ofx(t). Plot the magnitude spectrum corresponding tox(t).
5.4. Fourier and Laplace transforms—MATLAB Consider the signalx(t)=2e−2tcos(2πt)u(t).
(a) Use the fact this signal is bounded by the exponential±2e−2tu(t)to show that the integral Z∞
−∞
|x(t)|dt
Problems 353
is finite, indicating the signal is absolutely integrable and also finite energy.
(b) Use the Laplace transform to find the Fourier transformX()ofx(t).
(c) Use the MATLAB functionfourierto compute the magnitude and phase spectrum ofX().
5.5. Fourier transform of causal signals
Any causal signalx(t)having a Laplace transform with poles in the open-lefts-plane (i.e., not including the jaxis) has, as we saw before, a region of convergence that includes thejaxis, and as such its Fourier transform can be found from its Laplace transform. Consider the following signals:
x1(t)=e−2tu(t) x2(t)=r(t) x3(t)=x1(t)x2(t)
(a) Determine the Laplace transform of the above signals (use properties of the Laplace transform) indicating the corresponding region of convergence.
(b) Determine for which of these signals you can find its Fourier transform from its Laplace transform.
Explain.
(c) Give the Fourier transform of the signals that can be obtained from their Laplace transform.
5.6. Duality of Fourier transform
There are some signals for which the Fourier transforms cannot be found directly by either the integral definition or the Laplace transform, and for those we need to use the properties of the Fourier transform, in particular the duality property. Consider, for instance,
x(t)= sin(t) t
or the sinc signal. Its importance is that it is the impulse response of an ideal low-pass filter.
(a) LetX()=A[u(+0)−u(−0]be a possible Fourier transform ofx(t). Find the inverse Fourier transform ofX()using the integral equation to determine the values ofAand0.
(b) How could you use the duality property of the Fourier transform to obtainX()? Explain.
5.7. Cosine and sine transforms
The Fourier transforms of even and odd functions are very important. The reason is that they are computationally simpler than the Fourier transform. Letx(t)=e−|t|andy(t)=e−tu(t)−etu(−t).
(a) Plotx(t)andy(t), and determine whether they are even or odd.
(b) Show that the Fourier transform ofx(t)is found from X()=
Z∞
−∞
x(t)cos(t)dt
which is a real function of, thus its computational importance. Show thatX()is also even as a function of.
(c) FindX()from the above equation (called the cosine transform).
(d) Show that the Fourier transform ofy(t)is found from Y()= −j
Z∞
−∞
y(t)sin(t)dt
which is imaginary function of, thus its computational importance. Show thatY()is also odd as a function of.
(e) FindY()from the above equation (called the sine transform). Verify that your results are correct by finding the Fourier transform ofz(t)=x(t)+y(t)directly and using the above results.
(f) What advantages do you see to the cosine and sine transforms? How would you use the cosine and the sine transforms to compute the Fourier transform of any signal, not necessarily even or odd? Explain.
5.8. Time versus frequency—MATLAB
The supports in time and in frequency of a signal x(t) and its Fourier transform X() are inversely proportional. Consider a pulse
x(t)= 1
T0[u(t)−u(t−T0)]
(a) LetT0=1andT0=10and find and compare the corresponding|X()|.
(b) Use MATLAB to simulate the changes in the magnitude spectrum when T0=10k for k=0,. . ., 4 for x(t). Compute X() and plot its magnitude spectra for the increasing values of T0on the same plot. Explain the results.
5.9. Smoothness and frequency content—MATLAB
The smoothness of the signal determines the frequency content of its spectrum. Consider the signals x(t)=u(t+0.5)−u(t−0.5)
y(t)=(1+cos(πt))[u(t+0.5)−u(t−0.5)]
(a) Plot these signals. Can you tell which one is smoother?
(b) FindX()and carefully plot its magnitude|X()|versus frequency.
(c) Find Y() (use the Fourier transform properties) and carefully plot its magnitude |Y()| versus frequency.
(d) Which one of these two signals has higher frequencies? Can you now tell which of the signals is smoother? Use MATLAB to decide. Make x(t) andy(t)have unit energy. Plot 20 log10|Y()| and 20 log10|X()|using MATLAB and see which of the spectra shows lower frequencies.
5.10. Smoothness and frequency—MATLAB
Let the signalsx(t)=r(t+1)−2r(t)+r(t−1)andy(t)=dx(t)/dt.
(a) Plotx(t)andy(t).
(b) FindX()and carefully plot its magnitude spectrum. IsX()real? Explain.
(c) FindY()(use properties of Fourier transform) and carefully plot its magnitude spectrum. IsY()real?
Explain.
(d) Determine from the above spectra which of these two signals is smoother. Use MATLAB to plot 20 log10|Y()|and20 log10|X()|and decide. Would you say in general that computing the derivative of a signal generates high frequencies or possible discontinuities?
5.11. Integration and smoothing—MATLAB Consider the signal
x(t)=u(t+1)−2u(t)+u(t−1) and let
y(t)= Zt
−∞
x(τ)dτ
Problems 355
(a) Plotx(t)andy(t).
(b) FindX()and carefully plot its magnitude spectrum. IsX()real? Explain. (Use MATLAB to do the plotting.)
(c) FindY()and carefully plot its magnitude spectrum. IsY()real? Explain. (Use MATLAB to do the plotting.)
(d) Determine from the above spectra which of these two signals is smoother. Use MATLAB to decide.
Would you say that in general by integrating a signal you get rid of higher frequencies, or smooth out a signal?
5.12. Duality of Fourier transforms
As indicated by the derivative property, if we multiply a Fourier transform by (j)N, it corresponds to computing anNthderivative of its time signal. Consider the dual of this property—that is, if we compute the derivative ofX(), what would happen to the signal in the time?
(a) Letx(t)=δ(t−1)+δ(t+1). Find its Fourier transform (using properties)X().
(b) ComputedX()/dand determine its inverse Fourier transform.
5.13. Periodic functions in frequency
The duality property provides interesting results. Consider the signal x(t)=δ(t+T1)+δ(t−T1) (a) FindX()=F[x(t)]and plot bothx(t)andX().
(b) Suppose you then generate a signal
y(t)=δ(t)+ X∞ k=1
[δ(t+kT0)+δ(t−kT0)]
Find its Fourier transformY()and plot bothy(t)andY().
(c) Arey(t) and the corresponding Fourier transformY() periodic in time and in frequency? If so, determine their periods.
5.14. Sampling signal The sampling signal
δTs(t)= X∞
n=−∞
δ(t−nTs)
will be important in the sampling theory later on.
(a) As a periodic signal of periodTs, expressδTs(t)by its Fourier series.
(b) Determine then the Fourier transform1()=F[δTs(t)].
(c) PlotδTs(t)and1()and comment on the periodicity of these two functions.
5.15. Piecewise linear signals
The derivative property can be used to simplify the computation of some Fourier transforms. Let x(t)=r(t)−2r(t−1)+r(t−2)
(a) Find and plot the second derivative with respect totofx(t), ory(t)=d2x(t)/dt2. (b) FindX()fromY()using the derivative property.
(c) Verify the above result by computing the Fourier transformX()directly fromx(t)using the Laplace transform.
5.16. Periodic signal-equivalent representations
Applying the time and frequency shifts it is possible to get different but equivalent Fourier transforms of periodic signals. Assume a period of a periodic signalx(t)of periodT0isx1(t), so that
x(t)=X k
x1(t−kT0)
and as seen in Chapter 4 the Fourier series coefficients ofx(t)are found asXk=X1(jk0)/T0, so thatx(t) can also be represented as
x(t)= 1 T0
X
k
X1(jk0)ejk0t
(a) Find the Fourier transform of the first expression given above forx(t)using the time-shift property.
(b) Find the Fourier transform of the second expression forx(t)using the frequency-shift property.
(c) Compare the two expressions and comment on your results.
5.17. Modulation property
Consider the raised-cosine pulse
x(t)=[1+cos(πt)](u(t+1)−u(t−1))
(a) Carefully plotx(t).
(b) Find the Fourier transform of the pulsep(t)=u(t+1)−u(t−1).
(c) Use the definition of the pulsep(t)and the modulation property to find the Fourier transform ofx(t)in terms ofP()=F[p(t)].
5.18. Solution of differential equations
An analog averager is characterized by the relationship dy(t)
dt =0.5[x(t)−x(t−2)]
wherex(t)is the input andy(t)the output. Ifx(t)=u(t)−2u(t−1)+u(t−2): (a) Find the Fourier transform of the outputY().
(b) Findy(t)fromY(). 5.19. Generalized AM
Consider the following generalization of amplitude modulation where instead of multiplying by a cosine we multiply by a periodic signal with harmonic frequencies much higher than those of the message. Suppose the carrierc(t)is a periodic signal with fundamental frequency0, let’s say
c(t)= 6 X
k=4
2 cos(k0t)
and that the message is a sinusoid of frequency0=2π, orx(t)=cos(0t).
(a) Find the AM signals(t)=x(t)c(t). (b) Determine the Fourier transformS().
(c) What would be a possible advantage of this generalized AM system? Explain.
Problems 357
5.20. Filter for half-wave rectifier
Suppose you want to design a dc source using a half-wave rectified signalx(t)and an ideal filter. Letx(t) be periodic,T0=2, and with a period
x1(t)=
sin(πt) 0≤t≤1 0 1<t≤2,
(a) Find the Fourier transformX()ofx(t), and plot the magnitude spectrum including the dc and the first three harmonics.
(b) Determine the magnitude and cut-off frequency of an ideal low-pass filterH(j)such that when we havex(t)as its input, the output isy(t)=1. Plot the magnitude response of the ideal low-pass filter.
(For simplicity assume the phase is zero.) 5.21. Passive RLC filters—MATLAB
Consider an RLC series circuit with a voltage source vs(t). Let the values of the resistor, capacitor, and inductor be unity. Plot the poles and zeros and the corresponding frequency responses of the filters with the output the voltage across the
(a) Capacitor (b) Inductor
(c) Resistor
Indicate the type of filter obtained in each case. Use MATLAB to plot the poles and zeros, the magnitude, and the phase response of each of the filters obtained above.
5.22. AM modulation and demodulation
A pure tone x(t)=4 cos(1000t) is transmitted using an AM communication system with a carrier cos(10,000t). The output of the AM system is
y(t)=x(t)cos(10,000t)
At the receiver, to recoverx(t)the sent signaly(t)needs first to be separated from the thousands of other signals. This is done with a band-pass filter with a center frequency equal to the carrier frequency, and the output of this filter then needs to be demodulated.
(a) Consider an ideal band-pass filterH(j). Let its phase be zero. Determine its bandwidth, center frequency, and amplitude so we get as its output10y(t). Plot the spectrum of x(t), 10y(t), and the magnitude frequency response ofH(j).
(b) To demodulate10y(t), we multiply it bycos(10,000t). You need then to pass the resulting signal through an ideal low-pass filter to recover the original signalx(t). Plot the spectrum of
z(t)=10y(t)cos(10,000t)
and from it determine the frequency response of the low-pass filterG(j)needed to recoverx(t). Plot the magnitude response ofG(j).
5.23. Ideal low-pass filter—MATLAB
Consider an ideal low-pass filterH(s)with zero phase and magnitude response
|H(j)| =
1 −π ≤≤π 0 otherwise
(a) Find the impulse responseh(t)of the low-pass filter. Plot it and indicate whether this filter is a causal system or not.
(b) Suppose you wish to obtain a band-pass filterG(j)fromH(j). If the desired center frequency of
|G(j)|is5π, and its desired magnitude is1at the center frequency, how would you processh(t)to get the desired filter? Explain your procedure.
(c) Use symbolic MATLAB to findh(t),g(t), andG(j). Plot|H(j)|,h(t),g(t), and|G(j)|. 5.24. Magnitude response from poles and zeros—MATLAB
Consider the following filters with the given poles and zeros and dc constant:
H1(s): K=1poles p1= −1,p2,3= −1±jπ; zeros z1=1,z2,3=1±jπ H2(s): K=1poles p1= −1,p2,3= −1±jπ; zeros z1,3= ±jπ
H3(s): K=1poles p1= −1,p2,3= −1±jπ; zero z1=1
Use MATLAB to plot the magnitude responses of these filters and indicate the type of filters they are.
5.25. Different types of AM modulations—MATLAB Let the signal
m(t)=sin(2πt)[u(t)−u(t−1)]
be the message or input to different types of AM systems with the output the following signals. Carefully plotm(t)and the following outputs in0≤t≤1and their corresponding spectra using MATLAB. Let the sampling period beTs=0.001.
(a) y1(t)=m(t)cos(20πt) (b) y2(t)=[1+m(t)] cos(20πt) 5.26. Windows—MATLAB
The signalx(t)in Problem 5.17 is called a raised-cosine window. Notice that it is a very smooth signal and that it decreases at both ends. The rectangular window is the signaly(t)=u(t+1)−u(t−1).
(a) Use MATLAB to compute the magnitude spectrum ofx(t)andy(t)and indicate which is the smoother of the two by considering the presence of high frequencies as an indication of roughness.
(b) When computing the Fourier transform of a very long signal it makes sense to break it up into smaller sections and compute the Fourier transform of each. In such a case, windows are used to smooth out the transition from one section to the other. Consider a sinusoidz(t)=cos(2πt)for0≤t≤1000 sec.
Divide the signal into two sections of duration 500 sec. Multiply the corresponding signal in each of the sections by a raised-cosinex(t)and rectangulary(t)windows of length 500 and compute using MATLAB the corresponding Fourier transforms. Compare them to the Fourier transform of the whole signal and comment on your results. Sample all the signals usingTs=1/(4π)as the sampling period.
(c) Consider the computation of the Fourier transform of the acoustic signal corresponding to a train whistle, which MATLAB provides as a sampled signal in “train.mat” using the discrete approximation of the Fourier transform. The frequency content of the whole signal (hard to find) would not be as meaningful as the frequency content of a smaller section of it as they change with time. Compute the Fourier transform of sections of 1000 samples by windowing the signal with the raised-cosine window (sampled with the same sampling period as the “train.mat” signal orTs=1/FswhereFsis the sampling frequency given for “train.mat”). Plot the spectra of a few of these segments and comment on the change in the frequency content as time changes.