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.

Solution



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.)

Solution



Problem 6.

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

a) x(n) = u(n)

b) x(n) = ()n , n 0
3n , elsewhere

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

Solution



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.

Solution



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 .

Solution



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,].

Solution



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?

Solution



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)?

Solution



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).

Solution



Problem 13.

Find the Fourier series of the 2-periodic signal

f(x) = x , kun 0 x
- x , kun < x < 2.

Solution



Problem 14.

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

Solution



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.

Solution



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).

Solution



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).

Solution



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.

Solution



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?

Solution



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 ?

Solution



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?

Solution



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).

Solution



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?

Solution



Problem 24.

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

Solution



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).

Solution



Problem 27.

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

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

Solution



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.

Solution



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.

Solution



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.

Solution



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.

Solution



Problem 32.

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

Solution



Problem 33.

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

Solution



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.

Solution



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.

Solution



Problem 38.

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

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

Solution



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.

Solution



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.

Solution



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?

Solution



Problem 43.

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

Solution



Problem 44.

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

Solution



Problem 45.

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

Solution



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

Solution



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?

Solution



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.

Solution



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.

Solution



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.

Solution



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.

Solution



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.

Solution



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 + e-j .
1 - e-j

Is the system stable?

Solution



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.

Solution



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.

Solution



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).

Solution



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.

Solution



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)| = and h(1) > 0.

Solution



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.

Solution



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.

Solution



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.

Solution



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}

Solution



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.

Solution



Problem 66.

Transform the analog low pass filter

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

a) approximation of the derivative

Solution

b) impulse-invariance

Solution

c) bilinear mapping

Solution



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.

Solution


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