6.
Matlab m-file:
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% Sinewave Generation
%
fsig = signal frequency
%
fsamp = sampling frequency
%
T time interval
%
%
%
% calculate time increment for sampling-rate fsamp in the interval [0,T)
t=0:1/fsamp:
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E2.5 Signals & Linear Systems
Tutorial Sheet 5 Laplace Transform & Frequency Response
(Lectures 7 - 9)
1.*
Using Laplace transform, solve the following differential equations:
a)
b)
y (0 ) y (0 ) 0 and f (t ) u (t )
= =
=
( D 2 + 4 D + 4) y (t ) = ( D + 1
E2.5 Signals & Linear Systems
Tutorial Sheet 1 SOLUTIONS
1. (i)
(ii)
(iii)
(iv)
(v)
Periodic with period 1. Odd because x(t ) = x(t ) .
A-periodic. Neither odd nor even.
A-periodic. Even because x(t ) = x(t ) .
Periodic with period 15, odd.
A-periodic, od
6.
Matlab m-script
%-DFT, windowing and zeropadding
% this scripts generate three plots representing the amplitude of a sine
% wave of a fixed frequency and fixed sampling rate, but changing the
% windwow size and with ot without zero-padding.
%Create sam
5.*
Using the initial and final value theorems, find the initial and final values of the zero-state response of a system
with the transfer function
H (s) =
6 s 2 + 3s + 10
2s 2 + 6s + 5
and the input is
a)
u (t )
b)
e t u (t ) .
6.* For a LTI system descr
6.* Find and sketch c(t ) = f1 (t ) * f 2 (t ) for the pairs of functions shown as follow:
a)
c)
b)
d)
7.* Find and sketch c(t ) = f (t ) * g (t ) for the pairs of functions shown below.
8. Matlab exercise. Write a routine in matlab that, given two functi
E2.5 Signals & Linear Systems
Tutorial Sheet 2 System Responses
1.
A Linear Time Invariant (LTI) system is specified by system equation
( D 2 + 4 D + 4) y (t ) =(t )
Df
a)
Find the characteristic polynomial, characteristic equation, characteristic roots a
E2.5 Signals & Linear Systems
Tutorial Sheet 4 Laplace Transform
(Support Lecture 6)
1.*
By direct integration, find the one-sided Laplace transforms of the following functions:
a)
b)
te t u (t )
c)
t cos 0t u (t ) .
d)
2.*
u (t ) u (t 1)
e 2t cos(5t + )
4.
Consider the rectangular function
1,
t < 1/ 2
(t ) = 1 / 2, t = 1 / 2
0,
otherwise
(i) Sketch x(t ) =
(ii) Sketch x(t ) =
1
(t k )
k =0
+
(t k ) . (Hint: there is a simple way to express this signal.)
k =
5.
Consider a discrete-time signal x[n] , f
E2.5 Signals & Linear Systems
Tutorial Sheet 3 Zero-state Responses & Convolution
(Support Lectures 4 & 5)
1.*
Using direct integration, find the expression for:
a)
b)
y (t ) = e at u (t ) * e bt u (t )
c)
2.*
y (t ) = u (t ) * u (t )
y (t ) = tu (t ) * u
E2.5 Signals & Linear Systems
Tutorial Sheet 7 Sampling
(Lectures 12 - 13)
1.*
By applying the Parsevals theorem, show that
2.*
sinc 2 (kx)dx = .
k
Fig. Q2 (a) and (b) shows Fourier spectra of signals f1 (t ) and f 2 (t ) . Determine the Nyquist sampling
sin[0 (t )] u (t )
e)
5.* Find the inverse Laplace transform of the function:
2 s + 5 2 s .
e
s 2 + 5s + 6
6.* The Laplace transform of a causal periodic signal can be found from the knowledge of the Laplace transform of
its first cycle alone.
a)
If the L
5.
Find the inverse z-transform of
(a)*
X [ z] =
z ( z 4)
z 5z + 6
(b)*
X [ z] =
z (e 2 2)
(e 2 2)( z 2)
(c)*
X [ z] =
z (5 z + 22)
( z + 1)( z 2) 2
2
6. MATLAB exercise. Using the m-file sinegen of the previous class problem. Compute and plot the amplitu
4.* For the signal e at u (t ) , determine the bandwidth of an anti-aliasing filter if the essential bandwidth of the signal
contains 99% of the signal energy.
5.* A zero-order hold circuit shown in Fig. Q5 is often used to reconstruct a signal f(t) from
E2.5 Signals & Linear Systems
Tutorial Sheet 8 DFT and z-transform
(Lectures 14 - 15)
1.*
For a signal f(t) that is time-limited to 10 ms and has an essential bandwidth of 10 kHz, determine N 0 , the
number of signal samples necessary to compute a power o
E2.5 Signals & Linear Systems
Tutorial Sheet 1 Introduction to Signals & Systems
(Lectures 1 & 2)
1.
Sketch each of the following continuous-time signals, specify if the signal is periodic/non-periodic, odd/even. If
the signal is periodic specify its peri
E2.5 Signals & Linear Systems
Tutorial Sheet 6 Fourier Transform
(Lectures 10 - 11)
1.*
Derive the Fourier transform of the signals f(t) shown in Fig. Q1 (a) and (b).
(a)
(b)
Figure Q1
2.*
Sketch the following functions:
a)
t
rect ( )
2
c)
sinc(
5
t 10
)
Fig. Q5
5.* The signals in Fig. Q6 (a)-(c) are modulated signals with carrier cos 10t. Find the Fourier transforms of these
signals using appropriate properties of the Fourier transform and the FT table given in Lecture 10, slides 13-15.
Sketch the amplit