Contents

#### Signal and Systems. Problems

Problem 1.

Let (x(n))n=02 = { 2, 1, 3} and (y(n))n=02 = { 3, 1, -2 }, x(n)=0=y(n) for other values of the index n.

Calculate the convolution signal x(n) * y(n).

Problem 2.

Let (x(n))n=04 = { 2, 1, 2, -1, 3} and (y(n))n=02 = { 2, 1, 2 }, x(n)=0=y(n) for other values of the index n.

Calculate the convolution signal x(n) * y(n). The length of x(n) is L=5 and the length of y(n) is M=3.
What is the length N of the convolution x(n) * y(n)?
Guess what is the relation of L,M and N, in general.

Problem 3.

Find the autocorrelation function of the signal x(n),   - < n < ,

x(n) = anu(n),   - 1 < a < 1.

Problem 4.

Determine the a) cross correlation function
b) convolution

of the signals

x(n) = {...,0,0,-2,1,0,1,0,0,...}, x(0) = -2,

and

y(n) = {...,0,0,1,1,1,1,1,0,0,...}, y(-1) = 0, y(0) = 1,

Problem 5.

In the Spread spectrum technique the so called m-sequences, which are N = (2m - 1) - periodic, are used. Determine the autocorrelation function of the 7-periodic sequence below.

··· + 1  - 1  + 1  - 1  - 1  + 1  + 1  + 1  - 1  + 1  - 1···

(Find it only for the values 0, 1 and 2 of the delay.)

Problem 6.

Find which of the following discrete signals are energy signals and which power signals

a) x(n) = u(n)

 b) x(n) = (½)n , n 0 3n , elsewhere

c) x(n) = Ae j0n, where A and 0 are constants.

Problem 7.

 Let (x) = 1, 0 x 1, 0, elsewhere.
The so called Haar-wavelet W is defined by

W(x) = (2x) - (2x - 1).

The functions

m,n(x) = 2-m/2W(2-mx - n),  m,n Z,  x R,

form an orthonormal wavelet basis of L2(R). Sketch the graphs of the basis functions m,n, m = -2, -1, 0, 1,  n = 0, 1.

Problem 8.

Let the coefficients of the enrgy signal f(x),   - < x < , in the wavelet basis of Problem 7 be

 (f |m,n ) = am,n = 2-n/2/(m + 1), m 0,  n 0 0, elsewhere.

Find the energy of the signal f .

Problem 9.

Analog signal x(t) = cos (2500t) is sampled at frequency fs = 1 kHz (T = 1ms = 10-3s). Determine the obtained discrete signal x(n), whose angle frequency
[0,].

Problem 10.

Find another analog signal cos (t) which produces, with the sampling frequency fs = 1 kHz (T = 1ms = 10-3s) , the same sample sequence (the same discrete signal) as cos (2500t) in Problem 1? Sketch the graphs of that analog signal and that of Ex.1. What can you deduce from this?

Problem 11. After learning the Sampling Theorem we know the answer to:

What is the smallest sampling frequency, in order that the signal x(t) = cos (2500t) can be (uniquely) reconstructed from the samples x(n)?

Problem 12.

a)

Analog signal x(t) = cos (2500t) is sampled with sampling frequency fs = 4 kHz. Determine the obtained discrete signal x(n) and its digital frequency. Compare the result with the result of Problem 1.

b)

Analog signal x(t) = cos (3500t) is sampled with sampling frequency fs = 4 kHz. Determine the obtained discrete signal x(n) and its digital frequency. Compare the result with the result of a).

Problem 13.

Find the Fourier series of the 2-periodic signal

 f(x) = x , kun 0 x - x , kun < x < 2.

Problem 14.

Show the time shift and time scaling properties of the Fourier transform.

Problem 15.

 Let (x) = 1, 0 x 1, 0, elsewhere.
Haar-wavelet W is defined by the equation

W(x) = (2x) - (2x - 1).

Find the W Fourier transform of the Haar-wavelet.

Problem 16.

Find the smallest sampling frequency, for which it is possible to reconstruct the signal x(t) from the samples x(n) when

 x(t) = 2 cos (100t) + 5 sin (250t + /3) - 4.5 cos (375t - /4) + 15.9 sin (600t + /9).

Problem 17.

The amplitude spectrum of an analog signal f(t) is shown in Figure 1 and that of the sample sequence taken from f(t) in Figure 2. What was the sampling interval? What is the critical sampling frequencys of this signal. Sketch the amplitude spectrum of the sampled sequence when the sampling interval is 1/8 (time units).

Problem 18.

Show that the inverse discrete Fourier transform

(k) e j2kn/N = f(n)

really gives back the original values f(n), n = 0, 1, ..., N - 1.

Problem 19.

An analog signal is sampled with sampling frequency 3 kHz for 0.1 seconds and the amplitude spectrum estimated by using DFT. What is the highest possible frequency in the spectrum and what is the resolution f of the spectrun?

Problem 20.

a) 100 samples with sampling interval 1) T = 0.01s, 2) T = 0.02s.
b) for 0.2 seconds 1) 100, 2) 200 samples
c) with 0.1 sampling interval 1) 100, 2) 200 samples
are taken from an analog signal x(t). Determine the resolution of the spectrum. What analog frequencies can be seen in the spectrum ?

Problem 21.

An analog signal, which is a sum of 4 sinusoidal signals, is sampled with 800 samples 0.001 seconds apart and an 800 point discrete Fourier transform is calculated resulting in an amplitude spectrum shown in figure below. What are the analog frequencies of the 4 sinusoidal signals in question?

Problem 22.

Let m(t) be a bandlimited signal, whose Fourier transform M(v) = 0, when |v| > 200Hz. Determine the smallest sampling frequency and the largest sampling interval T, for which the signal x(t) = m(t) + 5 cos(100t) + 3 cos(200t+/4) + sin(500t) can be reconstructed from the samples x(nT).

Problem 23.

If the signal x(t) of the Problem 22. is sampled with 400 samples 0.005 seconds apart and if a 50Hz frequency component is to be filtered out from the sampled signal, then what condition the freguency response H() should satisfy?

Problem 24.

Determine the matrices which can be used in performing the 4 point IDFT and DFT.

Problem 25.

The Fourier transform of the discrete signal x(n) is

 X() = 1, || /4 0, || > /4.

Determine x(n).

Problem 26.

The signal x(n) takes values x(0) = ¼, x(1) = ¼, x(2) = ¼, x(n) = 0 elsewhere. Determine and sketch the graphs of the amplitude spectrum and the phase spectrum of x(n).

Problem 27.

Find the Fourier transform, amplitude spectrum and the phase spectrum of the signal

 x(n) = A, 0 n L-1 0, elsewhere.

Problem 28.

Determine the Fourier transform, amplitude spectrum and the phase spectrum of the unit step signal

x(n) = u(n)

by using Z-transform tables in finding the Fourier transform.

Problem 29.

Determine the Fourier transform, amplitude spectrum and the phase spectrum of the unit step signal

x(n) = (-1)nu(n)

by using Z-transform tables in finding the Fourier transform.

Problem 30.

Determine the Fourier transform, amplitude spectrum and the phase spectrum of the unit step signal

x(n) = (cos 0 n) u(n)

by using Z-transform tables in finding the Fourier transform.

Problem 31.

The zero-order hold in reconstructing an analog signal from samples x(nT),

(t) = x(nT),  nT t < nT + T,

can be understood as an analog LTI system, whose impulse response is

 h(t) = 1, 0 t T 0, elsewhere.

Justify this statement.

Hint: The input is

f(t) = x(nT) (t - nT)

and the output is

y(t) = h(s) f(t-s) ds = f(s) h(t-s) ds.

Show that y(t) = (t), when h(t) is the impulse response given above.

Problem 32.

Determine the transfer function and the amplitude response of the zero order hold and sketch the graph of the amplitude response.

Problem 33.

Determine the phase response of the zero order hold and sketch its graph.

Problem 34.

An analog LTI system is defined by the differential equation

y'(t) + 2y(t) = x(t-2),  t 0,

with the initial condition y(0) = 0, x(t)=0, t 0, where x(t) is the input and y(t) is the output. Determine the amplitude response, impulse response and the phase response of this system. Sketch their graphs.

Problem 35.

The current y(t) in the RC circuit is given by the equation

 R y(t) + 1 y(s) ds + v0 = x(t),  t > 0. C

Let y(0) = y0 = 0 and v0 = 0. Determine the transfer function and the impulse response of the above LTI system.

Problem 36.

Determine the output y(n) of a causal discrete LTI system, whose impulse response is (h(n))n=02 = { 1, 3, 2 }, and the input is (x(n))n=02 = { 2, 1, 4 }, x(n)=0=h(n) for other values of the index n.

Problem 37.

Determine the band width of the zero order hold.

Problem 38.

Determine the band width of the n. order Butterworth-low pass filter

 |H(f)|2 = 1 1 + (f/B)2n

Problem 39.

The current y(t) of the RL circuit satisfies the differential equation

Ly'(t) + Ry(t) = x(t),  t 0,

with the initial condition y(0) = y0 = 0.

a) Determine the transfer function and the impulse response of the above LTI system.

Study whether the RL circuit is
b) stable,
c) asymptotically stable,
d) BIBO stable.

Problem 40.

The current y(t) in the RC circuit is given by the equation

 R y(t) + 1 y(s) ds + v0 = x(t),  t > 0. C

Study whether the RC circuit is
a) stable,
b) asymptotically stable,
c) BIBO stable.

Problem 41.

Problem 42.

Find the impulse response of the band pass filter

 H(f) = 1, f1 |f| f2 0, elsewhere

and draw its graph. Is this filter causal and if not, by what change of the transfer function it can be made almost causal?

Problem 43.

Determine the Hilbert transform of the signal x(t) = sin (2fc t).

Problem 44.

Determine the Hilbert transform x(t) = sinc t.

Problem 45.

Determine the complex envelope (t) of the signal x(t) = sinc t.

Problem 46.

Determine the pre envelope x+(t) and the complex envelope (t) of the RF-pulse

 x(t) = A rect ( t ) cos 2fct. T

Problem 47.

The signal x(t) = Ac cos 2fct + m(t) is the input of a square-law detector. Find the spectrum of the output y(t) = a1 x(t) + a2 x2(t), when

 M(v) = (v+B)/B, -B v 0 (-v+B)/B, 0 v B 0, elsewhere.

What kind of filter would extract m(t) from the output?

Problem 48.

Let m(t) be the square wave given in the figure below, which frequency modulates the carrier wave Ac cos (2fc t) the frequency sensitivity being kf Hz/volt. Determine the wave form of the instantaneous frequency of the obtained FM wave.

Problem 49.

Let m(t) be the square wave given in Problem 48, which frequency modulates the carrier wave Ac cos (2fc t) the frequency sensitivity being kf Hz/volt. Determine the wave form of the phase of the obtained FM wave.

Problem 50.

Let m(t) be the square wave given in Problem 48, which frequency modulates the carrier wave Ac cos (2fc t) the frequency sensitivity being kf Hz/volt. Determine the complex envelope of the obtained FM wave.

Problem 51.

The amplitude sectrum of the signal x(t) is given in the figure below. Find the smallest sampling frequency (minimum sampling rate) and the corresponding sampling interval to reconstruct x(t) from samples.

Problem 52.

The amplitude sectrum of the signal x(t) is given in the figure below. Find the smallest sampling frequency (minimum sampling rate) and the corresponding sampling interval to reconstruct x(t) from samples.

Problems 53.-54.

Problem 55.

Find the output y(n) of the LTI system if the input is

x(n) = cos (n/2 + /4),   - < n < ,

and when the frequency response is

 H() = 1 + ½ e-j . 1 - ½ e-j

Is the system stable?

Problem 56.

The LTI system is defined by the difference equation

y(n) = a y(n-1) + b x(n),  0 < a < 1.

Determine the frequency response, amplitude response and the phase response of the system.

Problem 57.

The LTI system is defined by the difference equation

y(n) = a y(n-1) + b x(n),  0 < a < 1.

Find such a b, that the maximum of the amplitude reponse is 1, and sketch the graphs of the amplitude response and the phase response.

Problem 58.

The LTI system is defined by the difference equation

y(n) = a y(n-1) + b x(n),  0 < a < 1.

Determine the output if the input is

x(n) = 5 + 10 sin (n/2) - 15 cos (n + /4).

Problem 59.

The LTI system is defined by the difference equation

y(n) = 3 y(n-1) - 2 y(n-2) + x(n) + x(n-1).

Determine the transfer function, impulse response and study whether the system is Bibo stable.

Problem 60.

Determine the frequency response H() of the linear phase FIR filter and the phase response () by using the symmetry condition h(n) = h(M-1-n), where M = 4, and find the impulse response if it is required that H(0) = 1, |H(/3)| = ½ and h(1) > 0.

Problem 61.

Determine the frequency response H() of the linear phase FIR filter and the phase response () by using the antisymmetry condition h(n) = - h(M-1-n), where M = 5, and find the impulse response h(n) if it is required that |H(/2)| = 1, |H(/4)| = 2/3 and h(1) > 0.

Problem 62.

Determine such a stable rational transfer function of an LTI system by placing zeros and poles, that the frequency reponse H() has the values H(/2) = 1, H(0) = H() = 0 and |H(/3)| = 1/4.

Problem 63.

Determine such a stable rational transfer function of an LTI system by placing zeros and poles, that the frequency reponse H() has the values H(/2) = 1, H(0) = H() = 1 and (|H(/4)|)2 = 1/5.

Problem 64.

a) The impulse response of a discrete LTI system is h(n) = (2/3)nu(n). Determine the impulse response of the inverse system.

b) Determine the impulse response of a causal LTI system, if the output is

(y(n))n4=0 = {2,5,9,8,6}

when the input is

(x(n))n2=0 = {1,2,2}

Problem 65.

The input of a causal LTI system is y(n) = { -1, -2, 0, 4, 5, 2 }, when the input is x(n) = { -1, 0, 1, 2 }. Find the impulse response. The values of y(0) and x(0) are printed in bold.

Problem 66.

Transform the analog low pass filter

 Ha(s) = 1 s + 1
into a digital filter by the method of

Problem 67.

Transform the analog integrator, Ha(s) = 1/s , into a digital LTI system by using the bilinear mapping
 s = 2 1 - z-1 . T 1 + z-1
Determine H(), |H()| and () and the corresponding difference equation.

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