6.
Matlab m-file:
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% Sinewave Generation
%
fsig = signal frequency
%
fsamp = sampling frequency
%
T time interval
%
%
%
% calculate time increment for
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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 an
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 )
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
% win
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
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 exe
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 poly
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
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 si
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
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
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
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 t
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 F
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
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-periodi
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
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 tab