Let (*x*(*n*))* _{n=0}^{2}* = { 2, 1, 3} and (

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

Let (*x*(*n*))* _{n=0}^{4}* = { 2, 1, 2, -1, 3} and (

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.

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

Determine the **a)** cross correlation function

**b)** convolution

of the signals

and

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

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

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 |

3^{n} , |
elsewhere | |

**c)** *x*(*n*)* = Ae ^{ j0n}*, where

Let (x) = |
1, | 0 x 1, |

0, | elsewhere. | |

The functions

form an orthonormal wavelet basis of *L ^{2}*(

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

(f |)_{m,n } = a_{m,n } = |
2(^{-n/2}/m + 1), |
m 0, n 0 |

0, | elsewhere. |

Find the energy of the signal *f* .

Analog signal *x*(*t*)* = *cos (2500*t*) is sampled
at frequency *f _{s }* = 1 kHz (

[0,].

Find another analog signal cos (*t*) which produces, with the
sampling frequency *f _{s }* = 1 kHz (

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

**a)**

Analog signal *x*(*t*)* = *cos (2500*t*) is sampled
with sampling frequency *f _{s }* = 4 kHz.
Determine the obtained discrete signal

**b)**

Analog signal *x*(*t*)* = *cos (3500*t*) is sampled
with sampling frequency *f _{s }* = 4 kHz.
Determine the obtained discrete signal

Find the Fourier series of the 2-periodic signal

f(x) = |
x , |
kun 0 x |

- x , |
kun < x < 2. | |

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

Let (x) = |
1, | 0 x 1, |

0, | elsewhere. | |

Find the *W* Fourier transform of the Haar-wavelet.

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). | |

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).

Show that the inverse discrete Fourier transform

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

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?

**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 ?

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?

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

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

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

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

X() = |
1, | || /4 | |

0, | || > /4. | ||

Determine *x*(*n*).

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*).

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

x(n) = |
A, |
0 n L-1 |

0, | elsewhere. | |

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.

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

*x*(*n*)* = *(-1)* ^{n}u*(

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

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

*x*(*n*)* = *(cos * _{0 } n*)

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

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

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

and the output is

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

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

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

An analog LTI system is defined by the differential equation

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.

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

R y(t) + |
1 | y(s) ds + v(_{0} = xt), t > 0. |

C | ||

Let *y*(0)* = y _{0 } = *0 and

Determine the output * y*(*n*) of a causal discrete LTI system, whose impulse response is
(*h*(*n*))* _{n=0}^{2}* = {

Determine the band width of the zero order hold.

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

|H(f)|^{2} = |
1 | |

1 + (f/B)^{2n} | ||

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

with the initial condition *y*(0)* = y _{0} = *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.

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

R y(t) + |
1 | y(s) ds + v(_{0} = xt), t > 0. |

C | ||

Study whether the *RC* circuit is

**a)** stable,

**b)** asymptotically stable,

**c)** BIBO stable.

Find the impulse response of the band pass filter

H(f) = |
1, | f |_{1 } f| f_{2 } |

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?

Determine the Hilbert transform of the signal *x*(*t*)* = *sin (2*f _{c }t*).

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

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

Determine the pre envelope *x _{+}*(

x(t) = A rect ( |
t |
) cos 2f_{c}t. |

T | ||

The signal *x*(*t*)* = A _{c } *cos 2

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?

Let *m*(*t*) be the square wave given in the figure below, which frequency modulates
the carrier wave *A _{c }*cos (2

Let *m*(*t*) be the square wave given in Problem 48, which frequency modulates
the carrier wave *A _{c }*cos (2

Let *m*(*t*) be the square wave given in Problem 48, which frequency modulates
the carrier wave *A _{c }*cos (2

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.

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.

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

and when the frequency response is

H() = |
1 + ½ e^{-j} |
. |

1 - ½ e^{-j} | ||

Is the system stable?

The LTI system is defined by the difference equation

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

The LTI system is defined by the difference equation

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.

The LTI system is defined by the difference equation

Determine the output if the input is

The LTI system is defined by the difference equation

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

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.

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.

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.

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.

**a)**
The impulse response of a discrete LTI system is *h*(*n*) = (2/3)* ^{n}u*(

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

when the input is

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.

Transform the analog low pass filter

H(_{a}s) = |
1 | |

s + 1 | ||

**a)** approximation of the derivative

Transform the analog integrator, *H _{a}*(

s = |
2 | 1 - z^{-1} |
. |

T |
1 + z^{-1} | ||

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