Vinay K. Ingle, John G. Proakis
Chapter 3
THE DISCRETE-TIME FOURIER ANALYSIS - all with Video Answers
Educators
Chapter Questions
Using the matrix-vector multiplication approach discussed in this chapter, write a MATLAB function to compute the DTFT of a finite-duration sequence. The format of the function should be
```
function [X] = dtft(x,n,w)
% Computes Discrete-time Fourier Transform
% [X] = dtft (x,n,w)
% X = DTFT values computed at w frequencies
% x = finite duration sequence over n
% n}=\mathrm{ sample position vector
% w = frequency location vector
```
Use this function to compute the DTFT $X\left(e^{j \omega}\right)$ of the following finite-duration sequences over $-\pi \leq \omega \leq \pi$. Plot DTFT magnitude and angle graphs in one figure window.
1. $x(n)=(0.6)^{|n|}[u(n+10)-u(n-11)]$. Comment on the angle plot.
2. $x(n)=n(0.9)^n[u(n)-u(n-21)]$.
3. $x(n)=[\cos (0.5 \pi n)+j \sin (0.5 \pi n)][u(n)-u(n-51)]$. Comment on the magnitude plot.
4. $x(n)=\{4,3,2,1,1,2,3,4\}$. Comment on the angle plot.
5. $x(n)=\{4,3,2,1,-1,-2,-3,-4\}$. Comment on the angle plot.
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Let $x_1(n)=\{1,2,2,1\}$. A new sequence $x_2(n)$ is formed using
$$
x_2(n)= \begin{cases}x_1(n), & 0 \leq n \leq 3 ; \\ x_1(n-4), & 4 \leq n \leq 7 ; \\ 0, & \text { Otherwise. }\end{cases}
$$
1. Express $X_2\left(e^{j \omega}\right)$ in terms of $X_1\left(e^{j \omega}\right)$ without explicitly computing $X_1\left(e^{j \omega}\right)$.
2. Verify your result using MATLAB by computing and plotting magnitudes of the respective DTFTs.
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Determine analytically the DTFT of each of the following sequences. Plot the magnitude and angle of $X\left(e^{j \omega}\right)$ over $0 \leq \omega \leq \pi$.
1. $x(n)=2(0.5)^n u(n+2)$.
2. $x(n)=(0.6)^{|n|}[u(n+10)-u(n-11)]$.
3. $x(n)=n(0.9)^n u(n+3)$.
4. $x(n)=(n+3)(0.8)^{n-1} u(n-2)$.
5. $x(n)=4(-0.7)^n \cos (0.25 \pi n) u(n)$.
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The following finite-duration sequences are called windows and are very useful in DSP.
$$
\begin{aligned}
\text { Rectangular: } \mathcal{R}_M(n) & = \begin{cases}1,0 \leq n<M \\
0, & \text { otherwise }\end{cases} \\
\text { Hanning: } \mathcal{C}_M(n) & =0.5\left[1-\cos \frac{2 \pi n}{M-1}\right] \mathcal{R}_M(n) \\
\text { Triangular: } \mathcal{T}_M(n) & =\left[1-\frac{|M-1-2 n|}{M-1}\right] \mathcal{R}_M(n) ; \\
\text { Hamming: } \mathcal{H}_M(n) & =\left[0.54-0.46 \cos \frac{2 \pi n}{M-1}\right] \mathcal{R}_M(n)
\end{aligned}
$$
For each of these windows, determine their DTFTs for $M=10,25,50,101$. Scale transform values so that the maximum value is equal to 1 . Plot the magnitude of the normalized DTFT over $-\pi \leq \omega \leq \pi$. Study these plots and comment on their behavior as a function of $M$.
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Using the definition of the DTFT in (3.1), determine the sequences corresponding to the following DTFTs:
1. $X\left(e^{j \omega}\right)=3+2 \cos (\omega)+4 \cos (2 \omega)$.
2. $X\left(e^{j \omega}\right)=[1-6 \cos (3 \omega)+8 \cos (5 \omega)] e^{-\jmath 3 \omega}$.
3. $X\left(e^{j \omega}\right)=2+j 4 \sin (2 \omega)-5 \cos (4 \omega)$.
4. $X\left(e^{j \omega}\right)=[1+2 \cos (\omega)+3 \cos (2 \omega)] \cos (\omega / 2) e^{-j 5 \omega / 2}$.
5. $X\left(e^{j \omega}\right)=j[3+2 \cos (\omega)+4 \cos (2 \omega)] \sin (\omega) e^{-j 3 \omega}$.
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Using the definition of the inverse DTFT in (3.2), determine the sequences corresponding to the following DTFTs:
1. $X\left(e^{j \omega}\right)= \begin{cases}1, & 0 \leq|\omega| \leq \pi / 3 \\ 0, & \pi / 3<|\omega| \leq \pi\end{cases}$
2. $X\left(e^{j \omega}\right)= \begin{cases}0, & 0 \leq|\omega| \leq 3 \pi / 4 ; \\ 1, & 3 \pi / 4<|\omega| \leq \pi\end{cases}$
3. $X\left(e^{j \omega}\right)= \begin{cases}2, & 0 \leq|\omega| \leq \pi / 8 ; \\ 1, & \pi / 8<|\omega| \leq 3 \pi / 4 . \\ 0, & 3 \pi / 4<|\omega| \leq \pi .\end{cases}$
4. $X\left(e^{j \omega}\right)= \begin{cases}0, & -\pi \leq|\omega|<\pi / 4 ; \\ 1, & \pi / 4 \leq|\omega| \leq 3 \pi / 4 . \\ 0, & 3 \pi / 4<|\omega| \leq \pi .\end{cases}$
5. $X\left(e^{j \omega}\right)=\omega e^{j(\pi / 2-10 \omega)}$.
Remember that the above transforms are periodic in $\omega$ with period equal to $2 \pi$. Hence, functions are given only over the primary period of $-\pi \leq \omega \leq \pi$.
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A complex-valued sequence $x(n)$ can be decomposed into a conjugate symmetric part $x_e(n)$ and an conjugate anti-symmetric part $x_o(n)$ as discussed in Chapter 2. Show that
$$
\mathcal{F}\left[x_e(n)\right]=X_R\left(e^{j \omega}\right) \quad \text { and } \quad \mathcal{F}\left[x_o(n)\right]=j X_I\left(e^{j \omega}\right)
$$
where $X_R\left(e^{j \omega}\right)$ and $X_R\left(e^{j \omega}\right)$ are the real and imaginary parts of the DTFT $X\left(e^{j \omega}\right)$ respectively. Verify this property on
$$
x(n)=2(0.9)^{-n}[\cos (0.1 \pi n)+j \sin (0.9 \pi n)][u(n)-u(n-10)]
$$
using the MATLAB functions developed in Chapter 2.
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A complex-valued DTFT $X\left(e^{j \omega}\right)$ can also be decomposed into its conjugate symmetric part $X_c\left(e^{j \omega}\right)$ and conjugate anti-symmetric part $X_o\left(e^{j \omega}\right)$, i.e.,
$$
X\left(e^{j \omega}\right)=X_c\left(e^{j \omega}\right)+X_o\left(e^{j \omega}\right)
$$
where
$$
X_c\left(e^{j \omega}\right)=\frac{1}{2}\left[X\left(e^{j \omega}\right)+X^*\left(e^{-j \omega}\right)\right] \quad \text { and } \quad X_0\left(e^{j \omega}\right)=\frac{1}{2}\left[X\left(e^{j \omega}\right)-X^*\left(e^{-j \omega}\right)\right]
$$
Show that
$$
\mathcal{F}^{-1}\left[X_c\left(e^{j \omega}\right)\right]=x_R(n) \quad \text { and } \quad \mathcal{F}^{-1}\left[X_0\left(e^{j \omega}\right)\right]=j x_I(n)
$$
where $x_R(n)$ and $x_I(n)$ are the real and imaginary parts of $x(n)$. Verify this property on
$$
x(n)=e^{j 0.1 \pi n}[u(n)-u(n-20)]
$$
using the MATLAB functions developed in Chapter 2.
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Using the frequency-shifting property of the DTFT, show that the real part of $X\left(e^{j \omega}\right)$ of a sinusoidal pulse
$$
x(n)=\left(\cos \omega_o n\right) \mathcal{R}_M(n)
$$
where $\mathcal{R}_M(n)$ is the rectangular pulse given in Problem P3.4 is given by
$$
\begin{aligned}
X_{\mathrm{R}}\left(e^{j \omega}\right)= & \frac{1}{2} \cos \left\{\frac{\left(\omega-\omega_0\right)(M-1)}{2}\right\} \frac{\sin \left\{\left(\omega-\omega_0\right) M / 2\right\}}{\sin \left\{\left(\omega-\omega_0\right) / 2\right\}} \\
& +\frac{1}{2} \cos \left\{\frac{\left(\omega+\omega_0\right)(M-1)}{2}\right\} \frac{\sin \left\{\left[\omega-\left(2 \pi-\omega_0\right)\right] M / 2\right\}}{\sin \left\{\left[\omega-\left(2 \pi-\omega_0\right)\right] / 2\right\}}
\end{aligned}
$$
Compute and plot $X_{\mathrm{R}}\left(e^{j \omega}\right)$ for $\omega_0=\pi / 2$ and $M=5,15,25,100$. Use the plotting interval $[-\pi, \pi]$. Comment on your results.
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Let $x(n)=\mathcal{T}_{10}(n)$ be a triangular pulse given in Problem P3.4. Using properties of the DTFT, determine and plot the DTFT of the following sequences.
1. $x(n)=\mathcal{T}_{10}(-n)$
2. $x(n)=\mathcal{T}_{10}(n)-\mathcal{T}_{10}(n-10)$
3. $x(n)=\mathcal{T}_{10}(n) * \mathcal{T}_{10}(-n)$
4. $x(n)=\mathcal{T}_{10}(n) e^{j \pi n}$
5. $x(n)=\cos (0.1 \pi n) \mathcal{T}_{10}(n)$
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For each of the linear, shift-invariant systems described by the impulse response, determine the frequency response function $H\left(e^{j \omega}\right)$. Plot the magnitude response $\left|H\left(e^{j \omega}\right)\right|$ and the phase response $\angle H\left(e^{j \omega}\right)$ over the interval $[-\pi, \pi]$.
1. $h(n)=(0.9)^{|n|}$
2. $h(n)=\operatorname{sinc}(0.2 n)[u(n+20)-u(n-20)]$, where sinc $0=1$.
3. $h(n)=\operatorname{sinc}(0.2 n)[u(n)-u(n-40)]$
4. $h(n)=\left[(0.5)^n+(0.4)^n\right] u(n)$
5. $h(n)=(0.5)^{|n|} \cos (0.1 \pi n)$
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Let $x(n)=A \cos \left(\omega_0 n+\theta_0\right)$ be an input sequence to an LTI system described by the impulse response $h(n)$. Show that the output sequence $y(n)$ is given by
$$
y(n)=A\left|H\left(e^{j \omega_0}\right)\right| \cos \left[\omega_0 n+\theta_0+\angle H\left(e^{j \omega_0}\right)\right]
$$
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Let $x(n)=3 \cos \left(0.5 \pi n+60^{\circ}\right)+2 \sin (0.3 \pi n)$ be the input to each of the systems described in Problem P3.11. In each case, determine the output sequence $y(n)$.
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An ideal lowpass filter is described in the frequency domain by
$$
H_d\left(e^{j \omega}\right)=\left\{\begin{array}{cc}
1 \cdot e^{-j \alpha \omega}, & |\omega| \leq \omega_c \\
0, & \omega_c<|\omega| \leq \pi
\end{array}\right.
$$
where $\omega_c$ is called the cutoff frequency and $\alpha$ is called the phase delay.
1. Determine the ideal impulse response $h_d(n)$ using the IDTFT relation (3.2).
2. Determine and plot the truncated impulse response
$$
h(n)=\left\{\begin{array}{cc}
h_d(n), & 0 \leq n \leq N-1 \\
0, & \text { otherwise }
\end{array}\right.
$$
for $N=41, \alpha=20$, and $\omega_c=0.5 \pi$.
3. Determine and plot the frequency response function $H\left(e^{j \omega}\right)$, and compare it with the ideal lowpass filter response $H_d\left(e^{j \omega}\right)$. Comment on your observations.
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An ideal highpass filter is described in the frequency-domain by
$$
H_d\left(e^{j \omega}\right)=\left\{\begin{array}{cc}
1 \cdot e^{-j \alpha \omega}, & \omega_c<|\omega| \leq \pi \\
0, & |\omega| \leq \omega_c
\end{array}\right.
$$
where $\omega_c$ is called the cutoff frequency and $\alpha$ is called the phase delay.
1. Determine the ideal impulse response $h_d(n)$ using the IDTFT relation (3.2).
2. Determine and plot the truncated impulse response
$$
h(n)=\left\{\begin{array}{cc}
h_d(n), & 0 \leq n \leq N-1 \\
0, & \text { otherwise }
\end{array}\right.
$$
for $N=31, \alpha=15$, and $\omega_c=0.5 \pi$.
3. Determine and plot the frequency response function $H\left(e^{j \omega}\right)$, and compare it with the ideal highpass filter response $H_d\left(e^{j \omega}\right)$. Comment on your observations
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For a linear, shift-invariant system described by the difference equation
$$
y(n)=\sum_{m=0}^M b_m x(n-m)-\sum_{\ell=1}^N a_{\ell} y(n-\ell)
$$
the frequency-response function is given by
$$
H\left(e^{j \omega}\right)=\frac{\sum_{m=0}^M b_m e^{-j \omega m}}{1+\sum_{\ell=1}^N a_{\ell} e^{-j \omega \ell}}
$$
Write a MATLAB function freqresp to implement this relation. The format of this function should be
function [H] = freqresp(b,a,w)
% Frequency response function from difference equation
% [H] = freqresp(b,a,w)
%H= frequency response array evaluated at w frequencies
%b= numerator coefficient array
%a= denominator coefficient array (a(1)=1)
%w= frequency location array
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Determine $H\left(e^{j \omega}\right)$, and plot its magnitude and phase for each of the following systems:
1. $y(n)=\frac{1}{5} \sum_{m=0}^4 x(n-m)$
2. $y(n)=x(n)-x(n-2)+0.95 y(n-1)-0.9025 y(n-2)$
3. $y(n)=x(n)-x(n-1)+x(n-2)+0.95 y(n-1)-0.9025 y(n-2)$
4. $y(n)=x(n)-1.7678 x(n-1)+1.5625 x(n-2)+1.1314 y(n-1)-0.64 y(n-2)$
5. $y(n)=x(n)-\sum_{\ell=1}^5(0.5)^{\ell} y(n-\ell)$
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A linear, shift-invariant system is described by the difference equation
$$
y(n)=\sum_{m=0}^3 x(n-2 m)-\sum_{\ell=1}^3(0.81)^{\ell} y(n-2 \ell)
$$
Determine the steady-state response of the system to the following inputs:
1. $x(n)=5+10(-1)^n$
2. $x(n)=1+\cos (0.5 \pi n+\pi / 2)$
3. $x(n)=2 \sin (\pi n / 4)+3 \cos (3 \pi n / 4)$
4. $x(n)=\sum_{k=0}^5(k+1) \cos (\pi k n / 4)$
5. $x(n)=\cos (\pi n)$
In each case, generate $x(n), 0 \leq n \leq 200$, and process it through the filter function to obtain $y(n)$. Compare your $y(n)$ with the steady-state responses in each case.
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An analog signal $x_a(t)=\sin (1000 \pi t)$ is sampled using the following sampling intervals. In each case, plot the spectrum of the resulting discrete-time signal.
1. $T_s=0.1 \mathrm{~ms}$
2. $T_s=1 \mathrm{~ms}$
3. $T_s=0.01 \mathrm{sec}$
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We implement the following analog filter using a discrete filter.
$$
x_a(t) \longrightarrow \mathrm{A} / \mathrm{D} \xrightarrow{x(n)} h(n) \xrightarrow{y(n)} \mathrm{D} / \mathrm{A} \rightarrow y_a(t)
$$
The sampling rate in the $\mathrm{A} / \mathrm{D}$ and $\mathrm{D} / \mathrm{A}$ is $8000 \mathrm{sam} / \mathrm{sec}$, and the impulse response is $h(n)=(-0.9)^n u(n)$.
1. What is the digital frequency in $x(n)$ if $x_a(t)=10 \cos (10,000 \pi t)$ ?
2. Determine the steady-state output $y_a(t)$ if $x_a(t)=10 \cos (10,000 \pi t)$.
3. Determine the steady-state output $y_a(t)$ if $x_a(t)=5 \sin (8,000 \pi t)$.
4. Find two other analog signals $x_a(t)$, with different analog frequencies, that will give the same steady-state output $y_a(t)$ when $x_a(t)=10 \cos (10,000 \pi t)$ is applied.
5. To prevent aliasing, a prefilter would be required to process $x_\alpha(t)$ before it passes to the $\mathrm{A} / \mathrm{D}$ converter. What type of filter should be used, and what should be the largest cutoff frequency that would work for the given configuration?
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Consider an analog signal $x_a(t)=\cos (20 \pi t), 0 \leq t \leq 1$. It is sampled at $T_s=0.01,0.05$, and 0.1 sec intervals to obtain $x(n)$.
1. For each $T_s$ plot $x(n)$.
2. Reconstruct the analog signal $y_a(t)$ from the samples $x(n)$ using the sinc interpolation (use $\Delta t=0.001$ ) and determine the frequency in $y_a(t)$ from your plot. (Ignore the end effects.)
3. Reconstruct the analog signal $y_a(t)$ from the samples $x(n)$ using the cubic spline interpolation, and determine the frequency in $y_a(t)$ from your plot. (Again, ignore the end effects.)
4. Comment on your results.
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Consider the analog signal $x_a(t)=\cos (20 \pi t+\theta), 0 \leq t \leq 1$. It is sampled at $T_s=0.05$ sec intervals to obtain $x(n)$. Let $\theta=0, \pi / 6, \pi / 4, \pi / 3, \pi / 2$. For each of these $\theta$ values, perform the following.
1. Plot $x_a(t)$ and superimpose $x(n)$ on it using the plot $(\mathrm{n}, \mathrm{x}$, 'o') function.
2. Reconstruct the analog signal $y_a(t)$ from the samples $x(n)$ using the sinc interpolation (Use $\Delta t=0.001$ ) and superimpose $x(n)$ on it.
3. Reconstruct the analog signal $y_a(t)$ from the samples $x(n)$ using the cubic spline interpolation and superimpose $x(n)$ on it.
4. You should observe that the resultant reconstruction in each case has the correct frequency but a different amplitude. Explain this observation. Comment on the role of phase of $x_a(t)$ on the sampling and reconstruction of signals.
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