Solved ECE113: DSP Homework 1

Description

Problem 1:

1.1 Classify the following signals according to whether they are (1) one- or multi-dimensional; (2) single or multichannel, (3) continuous time or discrete time, and (4) analog or digital (in amplitude). Give a brief explanation.
(a) Closing prices of utility stocks on the New York Stock Exchange.
(b) A color movie.
(c) Position of the steering wheel of a car in motion relative to car’s reference frame.
(d) Position of the steering wheel of a car in motion relative to ground reference frame.
(e) Weight and height measurements of a child taken every month.

Problem 2:

1.3 Determine whether or not each of the following signals is periodic. In case a signal is periodic, specify its fundamental period.
(a) xa(t)=3cos⁡(5t+π/6)xa​(t)=3cos(5t+π/6)
(b) $x(n)=3\cos (5n+\pi /6)$
(c) $x(n)=2\exp [j(\pi /6-\pi )]$
(d) $x(n)=\cos (\pi /8)\cos (\pi n/8)$
(e) $x(n)=\cos (\pi n/2)-\sin (\pi n/8)+3\cos (\pi n/4+\pi /3)$

Problem 3:

1.5 Consider the following analog sinusoidal signal:

xa(t)=3sin⁡(100πt)xa​(t)=3sin(100πt)

(a) Sketch the signal xa(t)xa​(t) for $0\le t\le 30\text{ms}$.
(b) The signal xa(t)xa​(t) is sampled with a sampling rate Fs=300 samples/sFs​=300samples/s. Determine the frequency of the discrete-time signal x(n)=xa(nT)x(n)=xa​(nT), T=1/FsT=1/Fs​, and show that it is periodic.
(c) Compute the sample values in one period of $x(n)$. Sketch $x(n)$ on the same diagram with xa(t)xa​(t). What is the period of the discrete-time signal in milliseconds?
(d) Can you find a sampling rate FsFs​ such that the signal $x(n)$ reaches its peak value of 3? What is the minimum FsFs​ suitable for this task?

Problem 4:

2.1 A discrete-time signal $x(n)$ is defined as

x(n)={1+n3,−3≤n≤−11,0≤n≤30,elsewherex(n)=⎩⎨⎧​1+3n​,1,0,​−3≤n≤−10≤n≤3elsewhere​

(a) Determine its values and sketch the signal $x(n)$.
(b) Sketch the signals that result if we:
(1) First fold $x(n)$ and then delay the resulting signal by four samples.
(2) First delay $x(n)$ by four samples and then fold the resulting signal.
(c) Sketch the signal $x(-n+4)$.
(d) Compare the results in parts (b) and (c) and derive a rule for obtaining the signal $x(-n+k)$ from $x(n)$.
(e) Can you express the signal $x(n)$ in terms of signals $\delta (n)$ and $u(n)$?

Problem 5:

2.5 Show that the energy (power) of a real-valued energy (power) signal is equal to the sum of the energies (powers) of its even and odd components.

Problem 6:

2.10 The following input-output pairs have been observed during the operation of a time-invariant system:

x1(n)=[1,0,2]→Ty1(n)=[0,1,2]x1​(n)=[1,0,2]T​y1​(n)=[0,1,2]$$\uparrow$$x2(n)=[0,0,3]→Ty2(n)=[0,1,0,2]x2​(n)=[0,0,3]T​y2​(n)=[0,1,0,2]$$\uparrow$$x3(n)=[0,0,0,1]→Ty3(n)=[1,2,1]x3​(n)=[0,0,0,1]T​y3​(n)=[1,2,1]$$\uparrow$$

Can you draw any conclusions regarding the linearity of the system? What is the impulse response of the system?

Problem 7:

2.21 Compute the convolution $y(n)=x(n)\ast h(n)$ of the following pairs of signals.
(a) x(n)=anu(n),h(n)=bnu(n)x(n)=anu(n),h(n)=bnu(n) when a≠ba=b and when $a=b$

x(n)={1,n=−2,0,12,n=−10,elsewherex(n)=⎩⎨⎧​1,2,0,​n=−2,0,1n=−1elsewhere​$$h(n)=\delta (n)-\delta (n-1)+\delta (n-4)+\delta (n-5)$$

Problem 8:

2.23 Express the output $y(n)$ of a linear time-invariant (LTI) system with impulse response $h(n)$ in terms of its step response $s(n)=h(n)\ast u(n)$ and the input $x(n)$.

Problem 9:

1. (a) Let $x[n]$ and $y[n]$ be real-valued sequences both of which are even-symmetric: $x[n]=x[-n]$ and $y[n]=y[-n]$. Under these conditions, prove that rxy[ℓ]=ryx[ℓ]rxy​[ℓ]=ryx​[ℓ] for all $\ell$.

2. (b) Express the autocorrelation sequence rzz[ℓ]rzz​[ℓ] for the complex-valued signal $z[n]=x[n]+jy[n]$ where $x[n]$ and $y[n]$ are real-valued sequences, in terms of rxx[ℓ]rxx​[ℓ], rxy[ℓ]rxy​[ℓ], ryx[ℓ]ryx​[ℓ], and ryy[ℓ]ryy​[ℓ].

MATLAB Exercises:

Please submit your MATLAB script source code along with any necessary plots and discussion.

P2.8 The operation of signal dilation (or decimation or down-sampling) is defined by

$$y(n)=x(nM)$$

in which the sequence $x(n)$ is down-sampled by an integer factor $M$. For example, if

$$x(n)=\{\dots ,-2,4,3,-6,5,-1,8,\dots \}$$

then the down-sampled sequences by a factor 2 are given by

$$y(n)=\{\dots ,-2,3,5,8,\dots \}$$

1. Develop a MATLAB function dnsample that has the form

   matlab

   function [y,m] = dnsample(x,n,M)
   % Downsample sequence x(n) by a factor M to obtain y(m)

   to implement the above operation. Use the indexing mechanism of MATLAB with careful attention to the origin of the time axis $n=0$.

2. Generate $x(n)=\sin (0.125\pi n),-50\le n\le 50$. Decimate $x(n)$ by a factor of 4 to generate $y(n)$. Plot both $x(n)$ and $y(n)$ using subplot and comment on the results.

3. Repeat the above using $x(n)=\sin (0.5\pi n),-50\le n\le 50$. Qualitatively discuss the effect of down-sampling on signals.

P2.16 Let x(n)=(0.8)nu(n)x(n)=(0.8)nu(n), h(n)=(−0.9)nu(n)h(n)=(−0.9)nu(n), and $y(n)=h(n)\ast x(n)$. Use 3 columns and 1 row of subplots for the following parts.

1. Determine $y(n)$ analytically. Plot first 51 samples of $y(n)$ using the stem function.

2. Truncate $x(n)$ and $h(n)$ to 26 samples. Use conv function to compute $y(n)$. Plot $y(n)$ using the stem function. Compare your results with those of part 1.

3. Using the filter function, determine the first 51 samples of $x(n)\ast h(n)$. Plot $y(n)$ using the stem function. Compare your results with those of parts 1 and 2.