CHAPTER 10 Fourier Analysis of Discrete-Time Signals and Systems
10.5 What Have We Accomplished? Where Do We Go from Here?
In this chapter we have considered the Fourier representation of discrete-time signals and systems.
Just as with the Laplace and the Fourier transforms, in the continuous case there is a large class of discrete-time signals and impulse responses of systems for which we are able to find their discrete- time Fourier transform from their Z-transforms. For signals that are not absolutely summable, the time-frequency duality and other properties of the transform are used to find their DTFTs. Properties of the DTFT are very similar to those of the Z-transform. Although theoretically useful, the DTFT is computationally not feasible, due to the continuity of the frequency variable and to the integration required in the inverse transformation. It is the Fourier series of discrete-time signals that makes the Fourier representation computationally feasible. In Table 10.1, the DTFT of common signals and some DTFT properties are given.
The Fourier series coefficients constitute a periodic sequence of the same period as the signal; thus both are periodic. Moreover, the Fourier series and its coefficients are obtained as sums, and the frequency used is discretized. Thus, they can be obtained by computer. To take advantage of this, the spectrum of an aperiodic signal resulting from the DTFT is sampled so that in the time domain there is a periodic repetition of the original signal. For finite-support signals we can then obtain a periodic extension that gives the discrete Fourier transform or DFT. The significance of this result is that we have frequency representations of discrete-time signals that are computed algorithmically.
Table 10.2 displays properties of the discrete Fourier series and of the Discrete Fourier Transform (DFT). What remains then is to take a look at the algorithm used for those computations or the fast Fourier transform (FFT). We will do that in Chapter 12, where we will show that this algorithm efficiently computes the DFT and makes the convolution sum a more feasible procedure.
Problems 629
Table 10.1 DTFT of Common Signals and DTFT Properties
Discrete-Time Fourier Transforms
Discrete-Time Signal DTFTX(ejω), Periodic of Period2π
1. δ[n] 1, −π≤ω < π
2. A 2πAδ(ω), −π≤ω < π
3. ejω0n 2πδ(w−ω0), −π≤ω < π
4. αnu[n], |α|<1 1−α1e−jω, −π≤ω < π 5. nαnu[n], |α|<1 (1−αeαe−jω−jω)2, −π≤ω < π
6. cos(ω0n)u[n] π[δ(ω−ω0)+δ(ω+ω0)] , −π≤ω < π 7. sin(ω0n)u[n] −jπ[δ(ω−ω0)+δ(ω+ω0)] , −π≤ω < π
8. α|n|, |α|<1 1 1−α2
−2αcos(ω)+α2, −π≤ω < π
9. u[n+N/2]−u[n−N/2] sin(ω(sinN+1(ω/2))/2), −π≤ω < π
10. αncos(ω0n)u[n] 1 1−αcos(ω0)e−jω
−2αcos(ω0)e−jω+α2e−2jω, −π≤ω < π
11. αnsin(ω0n)u[n] 1 αsin(ω0)e−jω
−2αcos(ω0)e−jω+α2e−2jω, −π≤ω < π
Properties of the DTFT
Z-transform: x[n], X(z), |z| =1∈ROC X(ejω)=X(z)|z=ejω
Periodicity: x[n] X(ejω)=X(ej(ω+2πk)), kinteger Linearity: αx[n]+βy[n] αX(ejω)+βY(ejω)
Time-shift: x[n−N] e−jωNX(ejω) Frequency-shift: x[n]ejωon X(ej(ω−ω0))
Convolution: (x∗y)[n] X(ejω)Y(ejω)
Multiplication: x[n]y[n] 21πRπ
−πX(ejθ)Y(ej(ω−θ))dθ Symmetry: x[n], real valued |X(ejω)|, even function ofω
∠X(ejω),odd function ofω Parseval’s relation: P∞
n=∞|x[n]|2=21πRπ
−π|X(ejω|2dω
PROBLEMS
10.1. Eigenfunction property and frequency response—MATLAB An IIR filter is characterized by the difference equation
y[n]=0.5y[n−1]+x[n]−2x[n−1] n≥0
wherex[n]is the input andy[n]is the output of the filter. LetH(z)be the transfer function of the filter.
Table 10.2 Properties of Discrete Fourier Series and Discrete Fourier Transform.
Fourier Series of Discrete-Time Periodic Signals
x[n], periodic signal of periodN X[k], periodic FS coefficients of periodN Z-transform x1[n]=x[n](u[n]−u[n−N]) X[k]=N1Z(x1[n])
z=ej2πk/N
DTFT x[n]=P
kX[k]ej2πnk/N X(ejω)=P
k2πX[k]δ(ω−2πk/N) LTI response Input: x[n]=P
kX[k]ej2πnk/N Output: y[n]=P
kX[k]H(ejkω0)ej2πnk/N H(ejω), frequency response of system Time shift (circular shift) x[n−M] X[k]e−j2πkM/N
Modulation x[n]ej2πMn/N X[k−M]
Multiplication x[n]y[n] PN−1
m=0X[m]Y[k−m], periodic convolution Periodic convolution PN−1
m=0x[m]y[n−m] NX[k]Y[n]
Discrete Fourier Transform
x[n], finite-lengthNaperiodic signal x[n],˜ periodic extension of period L≥N
˜ x[n]=N1
PL−1
k=0X[k]e˜ j2πnk/L X[k]˜ =PL−1
n=0x[n]e˜ −j2πnk/L
IDFT/DFT x[n]= ˜x[n]W[k], W[n]=u[n]−u[n−N] X[k]= ˜X[k]W[k],W[k]=u[k]−u[k−N]
Circular convolution (x⊗Ly)[n] X[k]Y[k]
Circular and linear convolution (x⊗Ly)[n]=(x∗y)[n], L≥M+K−1 M=length ofx[n],K=length ofy[n]
(a) The given filter is LTI, and as such the eigenfunction property applies. Obtain the magnitude response|H(ejω)|of the filter using the eigenfunction property.
(b) Compute the magnitude response|H(ejω)|at discrete frequenciesω=0,π/2, andπradians. Show that the magnitude response is constant for0≤ω≤π, and as such this is an all-pass filter.
(c) Use the MATLAB functionfreqzto compute the frequency response (magnitude and phase) of this filter and to plot them.
(d) Determine the transfer functionH(z)=Y(z)/X(z). Find its pole and zero and indicate how they are related.
10.2. Frequency transformation of low-pass to high-pass filters—MATLAB
You have designed an IIR low-pass filter with an input–output relation given by the difference equation y[n]=0.5y[n−1]+x[n]+x[n−1] n≥0
wherex[n]is the input andy[n]is the output. You are told that by changing the difference equation to y[n]= −0.5y[n−1]+x[n]−x[n−1] n≥0
you obtain a high-pass filter.
Problems 631
(a) From the eigenfunction property find the frequency response of the two filters atω=0,π/2, andπ radians. Use the MATLAB functionsfreqzandabsto compute the magnitude responses of the two filters. Plot them to verify that the filters are low pass and high pass.
(b) CallH1(ejω)the frequency response of the first filter andH2(ejω) the frequency response of the second filter. Show that
H2(ejω)=H1(ej(π−ω)) and relate the impulse responseh2[n]toh1[n].
(c) Use the MATLAB functionzplaneto find and plot the poles and the zeros of the filters and determine the relation between the poles and the zeros of the two filters.
10.3. Computations from definition of DTFT and IDTFT
Consider the discrete-time signalx[n]=0.5|n|, and find its DTFTX(ejω). From the direct and the inverse DTFT ofx[n]:
(a) Determine the infinite sum
X∞
k=−∞
0.5|n|
(b) Find the integral
Zπ
−π
X(ejω)dω
(c) Find the phase ofX(ejω). (d) Determine the sum
X∞
k=−∞
(−1)n0.5|n|
10.4. Frequency shift of FIR filters—MATLAB
Consider a moving-average FIR filter with an impulse response h[n]= 1
3(δ[n]+δ[n−1]+δ[n−2]) LetH(z)be the Z-transform ofh[n].
(a) Find the frequency responseH(ejω)of the FIR filter.
(b) Let the impulse response of a new filter be given by h1[n]=h[n]ejπn
Use the eigenfunction property to find the frequency responseH1(ejω)of the new FIR filter.
(c) Use the MATLAB functionsfreqzandabsto compute the magnitude response of the two filters. Plot them and determine the location of the poles and the zeros of the two filters. What type of filters are these?
10.5. Duality of DTFT
The DTFT of a discrete-time signalx[n]is given as
X(ejω)=ejπ/4δ(ω−0.5π)+e−jπ/4δ(ω+0.5π)−2πe−jπ/8δ(ω−0.71)−2πejπ/8δ(ω+0.71) (a) Is the signalx[n]periodic? If so, indicate its period.
(b) Determine the signalx[n], and verify your answer above.
10.6. Chirps for jamming—MATLAB
A chirp signal is a sinusoid of continuously changing frequency. Chirps are frequently used to jam communication transmissions. Consider the chirp
x[n]=cos(θn2)u[n] θ= π
2L 0≤n≤L−1
(a) A measure of the frequency of the chirp is the so-called instantaneous frequency, which is defined as the derivative of the phase in the cosine—that is,
IF(n)= dθn2 dn
Find the instantaneous frequency of the given chirp. Use MATLAB to plotx[n]forL=256.
(b) LetL=256and use MATLAB to compute the DTFT ofx[n]and to plot its magnitude. Indicate the range of discrete frequencies that would be jammed by the given chirp.
10.7. Time specifications for FIR filters—MATLAB
When designing discrete filters the specifications can be given in the time domain. One can think of converting the frequency-domain specifications into the time domain. Assume you wish to obtain a filter that approximates an ideal low-pass filter with a cut-off frequencyωc=π/2and that has a linear phase
−Nω. Thus, the frequency response is H(ejω)=
1e−jNω −π/2≤ω≤π/2
0 −π≤ω < π/2 and π/2< ω≤π
(a) Find the corresponding impulse response using the inverse DTFT ofH(ejω).
(b) IfN=50, ploth[n]using the MATLAB functionstemfor0≤n≤100. Comment on the shape of the plot.
(c) Suppose we want a band-pass filter of center frequencyω0=π/2. Use the above impulse response h[n]to obtain the impulse response of the desired band-pass filter.
10.8. Z-transform and DTFT—MATLAB
Consider a discrete pulsep[n]=u[n]−u[n−N].
(a) Use the definition of the DTFT to determineP(ejω)and then use the Z-transformP(z)ofp[n]to verify your result.
(b) ForN=5,10, and20, use the MATLAB functionzplaneto find the zeros ofP(z)and indicate at what frequenciesP(ejω)is zero. Verify your response usingfreqz.
(c) Suppose the impulse response of a filter ish[n]=u[n]−u[n−4], and its input is
v[n]= 2 X k=1
cos(kω0n)
For what value ofω0is the steady-state response of the filter zero?
10.9. Downsampling and DTFT—MATLAB
Consider pulses x1[n]=u[n]−u[n−20] and x2[n]=u[n]−u[n−10], and their product x[n]= x1[n]x2[n].
(a) Plot the three pulses. Could you say thatx[n]is a downsampled version ofx1[n]? What would be the downsampling rate? FindX1(ejω).
(b) Find directly the DTFT ofx[n]and compare it toX1(ejω/M)whereMis the downsampling rate found above. If we downsamplex1[n]to getx[n], would the result be affected by aliasing? Use MATLAB to plot the magnitude DTFT ofx1[n]andx[n]to provide an answer.
Problems 633
10.10. Cascading of interpolators and decimators—MATLAB
Suppose you cascade an interpolator (an upsampler and a low-pass filter) and a decimator (a low-pass filter and a downsampler).
(a) If both the interpolator and the decimator have the same rateM, carefully draw a block diagram of the interpolator–decimator system.
(b) Suppose that the interpolator is of rate 3 and the decimator of rate 2. Carefully draw a block diagram of the interpolator–decimator system. What would be the equivalent of sampling the input of this system to obtain the same output?
(c) Use the MATLAB functionsinterpanddecimateto process the first 100 samples of the test signal handelwhere the interpolator’s rate is 3 and the decimator’s is 2. How many samples does the output have?
10.11. MATLAB and phase computation
The computation of the phase of a complex number or function using MATLAB has some issues that you need to understand:
(a) The range of possible values of the inverse tangent needs to be extended to[−π,π)depending on the quadrant the complex number is in. Consider the complex numbers1+j,−1+j,−1−j, and 1−j, and represent each by a vector from the origin and consider what changes are needed when we use the formula to find the phase.
(b) The phase of a complex number is only significant if its magnitude is significant. Use MATLAB to compute the magnitudes and the phases of the complex numbersx=1+jandy=10−6+j10−6. How do the phases of these numbers compare? What about their magnitudes? Explain.
(c) If the functionX(ejω)is zero or infinite at a frequencyω0, the phase is undetermined at that frequency and of no significance since the corresponding magnitude is zero or infinity. LetX(z)=z−1, so that X(ejω)=ejω−1. Can you determine the phase ofX(ej0)? Explain. Likewise, if X(z)=1/(z−1), what is the phase ofX(ejω)atω=0?
(d) If the phase is linear (i.e.,θ= −Nω), MATLAB will plot the values only between[−π,π]and so the phase will not appear linear. LetX(z)=z−4. Find the phase ofX(ejω)—is it linear? Then use the MATLAB functionsfreqzandangleto compute the phase ofX(ejω)—does it appear linear? Explain.
10.12. Linear phase and phase unwrapping—MATLAB
A DTFTX(ejω)is said to have linear phase if its phase is a line through the origin of the frequency plane. Let
X(ejω)=2e−j4ω −π≤ω < π
(a) Carefully plot the magnitude and the phase ofX(ejω). Is the phase linear?
(b) Use the MATLAB functionsfreqzandangleto compute the phase ofX(ejω)and then plot it. (Hint:
Letz=ejωto be able to usefreqz.) Does the phase computed by MATLAB appear linear? What are the maximum and minimum values of the phase, and how many radians separate the minimum from the maximum?
(c) Now, recalculate the phase, but after usingangleuse the functionunwrappingin the resulting phase and plot it. Does the phase appear linear?
10.13. Linear phase and symmetry—MATLAB
Consider the signalx[n]=Aδ[n]+u[n+9]−u[n−10].
(a) Carefully plotx[n]. Find the Z-transformX(z)ofx[n]and from itX(ejω), the DTFT ofx[n]. Find the value ofAso that the phase ofX(ejω)is zero. Use MATLAB to verify your results.
(b) Consider nowx1[n]=x[n−9], and use the value ofAfound before. Carefully plotx1[n]and find its DTFT using the Z-transformX1(z). Is its phase linear? Use MATLAB to verify your results. Usefreqz, angle, andunwrapto compute the phase.
10.14. Sinusoidal form of DTFT A triangular pulse is given by
t[n]=
3+n −2≤n≤ −1 3−n 0≤n≤2
0 otherwise
(a) The pulse can be written as
t[n]=
∞ X k=−∞
Akδ[n−k]
Find the{Ak}coefficients.
(b) Find a sinusoidal expression for the DTFT oft[n]—that is, T(ejω)=B0+
X∞
k=1
Bkcos(kω)
Express the coefficientsB0andBkin terms of theAkcoefficients.
10.15. DTFT and Z-transform—MATLAB
Letx[n]=r[n]−r[n−3]−u[n−3]wherer[n]is the ramp signal.
(a) Carefully plotx[n]and find its Z-transformX(z). (b) Ify[n]=x[−n], giveY(z)in terms ofX(z).
(c) Use the above results to find the DTFT ofx[n],x[−n], andx[n]+x[−n]. Find the magnitude of each of these DTFTs and then use MATLAB to compute them and plot them.
10.16. Computations from DTFT definition
For simple signals it is possible to obtain some information on their DTFTs without computing them. Let x[n]=δ[n]+2δ[n−1]+3δ[n−2]+2δ[n−3]+δ[n−4]
(a) FindX(ej0)andX(ejπ)without computing the DTFTX(ejω). (b) Find
Zπ
−π
|X(ejω)|2dω
(c) Find the phase ofX(ejω). Is it linear?
10.17. DTFT of even and odd functions A signal
x[n]=0.5nu[n]
is neither even nor odd.
(a) Find the evenxe[n]and the oddxo[n]components ofx[n], and carefully plot them.
(b) Find the Z-transforms ofxe[n]andxo[n], and from them find the DFTsXe(ejω)andXo(ejω). Are they real or imaginary?
(c) Sincex[n]=xe[n]+xo[n]so thatX(ejω)=Xe(ejω)+Xo(ejω), how dos the real and the imaginary parts ofX(ejω)relate toXe(ejω)andXo(ejω)? Explain.
(d) Use Parseval’s result to obtain thatEx=Exe+Exo i.e., the energy of the signal is the sum of the energies of its even and odd components.
Problems 635
10.18. Power spectral density
Consider an autocorrelation function
c[n]=0.5|n| − ∞<n<∞
(a) Find the magnitude square of the DTFTC(ejω)ofc[n], which is called the power spectral density.
(b) Find the Z-transform ofc[n]and determine where its poles and zeros are. Are there any zeros or poles on the unit circle?
(c) FindC(ej0)—that is, the dc value of the power spectral density. Determine the phase ofC(ejω)—is it linear?
10.19. Convolution sum and product of polynomials
The convolution sum can be seen as a way to compute the coefficients of the product of polynomials.
This is because
[x∗y][n]⇔X(z)Y(z)⇔X(ejω)Y(ejω)
(a) Let X(z)=1+2z−1+3z−2 and Y(z)=z−2+4z−3 if x[n]=1δ[n]+2δ[n−1]+3δ[n−2] and y[n]=1δ[n−2]+4δ[n−3]are sequences formed by the coefficients of the polynomials. Compute the convolution sum[x∗y][n]and compare it to the coefficients of the polynomialZ(z)=X(z)Y(z), orZ(ejω)=X(ejω)Y(ejω).
(b) Suppose that the transfer function of a discrete-time system is H(z)= W(z)
V(z) =3z2+2z+2z−1+3z−2
and that it is known that the input isv[n]=u[n]−u[n−3]. Use the connection between the product of the polynomials and the convolution sum to find the output w[n]of the system.
10.20. Windowing and DTFT—MATLAB
A window w[n]is used to consider the part of a signal we are interested in.
(a) Let w[n]=u[n]−u[n−20]be a rectangular window of length20. Letx[n]=sin(0.1πn). We are interested in a period of the infinite lengthx[n], ory[n]=x[n]w[n]. Compute the DTFT ofy[n]and compare it with the DTFT ofx[n]. Write a MATLAB script to computeY(ejω).
(b) Let w1[n]=(1+cos(2πn/11))(u[n+5]−un[n−5])be a raised-cosine window that is symmetric with respect ton=0(noncausal). Adapt the script in the previous part to find the DTFT of
z[n]=x[n]w1[n]
wherex[n]is the sinusoid given above.
10.21. Z-transform and Fourier series—MATLAB Let
x1[n]=0.5n 0≤n≤9
be a period of a periodic signalx[n].
(a) Use the Z-transform to compute the Fourier series coefficients.
(b) Use MATLAB to plot the magnitude and the phase line spectrum (i.e.,|Xk|and∠Xkversus frequency
−π≤ω≤π).
10.22. Linear equations and Fourier series—MATLAB
The Fourier series of a signalx[n]and its coefficientsXkare both periodic of some valueN, and as such can be written as
n x[n]= N−1
X k=0
Xkej2πnk/N 0≤n≤N−1
n Xk= 1 N
N−1 X n=0
x[n]e−j2πnk/N 0≤k≤N−1
(a) To find thex[n],0≤n≤N−1givenXk,0≤k≤N−1, write a set ofNlinear equations. Indicate how you would find thex[n]from the matrix equation.
(b) As you can see, there is a lot of duality in the Fourier series and its coefficients. If you consider the reverse problem in the previous part, how would you solve forXkgiven thex[n]?
(c) Letx[n]=nforn=0, 1, 2, and0forn=3be a period of a periodic signalx[n]of periodN=4. Use the above method to solve for the Fourier series coefficientsXk,0≤k≤3. Use MATLAB to find the inverse of the complex exponential matrix.
(d) Suppose that when computing theXkfor thex[n]signal given above, you separate the sum into two sums, one for the even values ofn(i.e.,n=0, 2) and the other for the odd values ofn(i.e.,n=1, 3).
Try to simplify the complex exponentials and write an equivalent matrix expression for theXk. 10.23. Operations on Fourier series—MATLAB
A periodic signalx[n]of periodNcan be represented by its Fourier series x[n]=
N−1 X k=0
Xkej2πnk/N 0≤n≤N−1
If you consider this a representation ofx[n]:
(a) Is x1[n]=x[n−3] periodic? If so, use the Fourier series of x[n] to obtain the Fourier series coefficients ofx1[n].
(b) Letx2[n]=x[n]−x[n−1](i.e., the finite difference). Determine ifx2[n]is periodic, and if so, find its Fourier series coefficients.
(c) Ifx3[n]=x[n](−1)n, isx3[n]periodic? If so, determine its Fourier series coefficients.
(d) Letx4[n]=sign[cos(0.5πn)]wheresign(ξ)is a function that gives1whenξ≥0and−1whenξ <0.
Determine the Fourier coefficients ofx4[n]if periodic.
(e) Use MATLAB to find the Fourier series coefficients forxi[n],i=1, 2, 3,and4and to plot them as functions ofk.
10.24. Fourier series of even and odd signals—MATLAB Letx[n]be an even signal andy[n]be an odd signal.
(a) Determine whether the Fourier coefficientsXk andYk corresponding tox[n]andy[n]are complex, real, or imaginary.
(b) Considerx[n]=cos(2πn/N)andy[n]=sin(2πn/N)forN=3andN=4. Use the above results to find the Fourier series coefficients for the two signals with the different periods.
(c) Use MATLAB to find the Fourier series coefficients of the above two signals with the different periods, and plot their magnitude and phase spectra.
10.25. Response of LTI systems to periodic signals—MATLAB Suppose you get noisy periodic measurements
y[n]=(−1)nx[n]+Aη[n]
Problems 637
wherex[n]is the desired signal andη[n]is a noise that varies from0to1at random.
(a) LetA=0andx[n]=sign[cos(0.1πn)]. Determine how to processy[n], indicating the type of filter to obtain an approximate version ofx[n]. Consider the first 100 samples ofx[n]and use MATLAB to find the spectrum ofx[n]andy[n]to show that the filter you recommend will do the job.
(b) Use the MATLAB functionfir1to generate the kind of filter you decided to use above and show that when filteringy[n]forA=0, you obtain the desired result.
(c) Consider the first 100 samples of the MATLAB file “handel.mat” a period of a signal that continuously replays these values over and over. Letx[n]be the desired signal that results from this. Now let A=0.01, and use the functionrandto generate the noise, and come up with suggestions as to how to get rid of the effects of the multiplication by(−1)n and of the noiseη[n]. Recover the desired signalx[n].
10.26. DFT of an aperiodic and periodic signal—MATLAB Consider a signalx[n]=(−0.95)n(u[n]−u[n−70]).
(a) To compute the DFT ofx[n]we pad it with zeros so as to obtain a signal with length2γ, the larger but closest to the length ofx[n]. Determine the value ofγand use the MATLAB functionfftto compute the DFTX[k]of the padded-with-zeros signal. Plot its magnitude and phase.
(b) Let nowx1[n]=x[n−10]. Compute its DFTX1[k]using thefftfunction. Padx1[n]with zeros to compute theγ-length FFT whereγ is the value obtained above.
(c) Considerx[n]a period of a periodic signal of periodN=70. Compute its DFT using thefftalgorithm and then plot its magnitude and phase. What is the length of the FFT? Can we pad with zeros the period to find its DFT?
10.27. Frequency resolution of DFT—MATLAB
When we pad an aperiodic signal with zeros, we are improving its frequency resolution—that is, the more zeros we attach to the original signal the better the frequency resolution, as we obtain the frequency representation at a larger number of frequencies around the unit circle.
(a) Consider an aperiodic signalx[n]=u[n]−u[n−10], and compute its DFT by means of thefftfunc- tion padding it with 10 and then 100 zeros. Plot the magnitude response usingstem. Comment on the frequency resolution of the two DFTs.
(b) When the signal is periodic, one cannot pad a period with zeros. When computing the FFT in theory we generate a periodic signal of periodLequal or larger than the length of the signal when the signal is aperiodic, but if the signal is periodic we must letLbe the signal period or a multiple of it. Adding zeros to the period makes the signal different from the periodic signal. Considerx[n]=cos(πn/5),
−∞<n<∞as a periodic signal, and do the following:
n Consider exactly one period ofx[n]and compute the FFT of this sequence.
n Consider 10 periods ofx[n]and compute the FFT of this sequence.
n Consider attaching 10 zeros to one period and compute the FFT of the resulting sequence.
If we consider the first of these cases giving the correct DFT ofx[n], how many harmonic frequencies does it show. What happens when we consider the 10 periods? Are the harmonic frequencies the same as before? What are the values of the DFT in frequencies in between the harmonic frequencies?
What happened to the magnitude at the original frequencies. Finally, does the last FFT relate at all to the first FFTs?
10.28. DFT and IIR filters—MATLAB
A definite advantage of the FFT is that it reduces considerably the computation in the convolution sum.
Thus, ifx[n],0≤n≤N−1 is the input of an FIR filter with impulse responseh[n], 0≤n≤M−1, their convolution sumy[n]=[x∗h][n]will be of lengthM+N−1. Now ifX[k]andH[k]are the DFTs (computed by the FFT) ofx[n]andh[n], and ifY[k]=X[k]H[k]is the DFT of the convolution sum of length bigger or equal toM+N−1, then to be able to multiply the FFTsX[k]andH[k]they both should