Signals and Systems Exam 2004-5
15/12/04
This document describes the range of material and type of questions that may
appear on the 2nd year Signals and Systems exam. The exam is 1.5 hours
long and you have to answer three from four questions. As the course is new,
there are no previous exam papers. The information below describes the
examinable material and gives exemplar questions.
The focus of the questions is on the understanding and application of the
results, rather than theoretical derivation, unless otherwise indicated below. If
the topic is not mentioned below, you may assume it is not examinable. The
exam will not cover any Matlab/Simulink.
You will have a formula book in the exam. In addition, basic formulae for the
Fourier/Laplace series and transforms and inverse transforms will be given,
where necessary, in the exam paper.
1. Introduction to Signals and Systems (Lectures 1-3)

Understand the form of basic continuous and discrete time signals,
including: impulse, step and exponential (real, imaginary and complex)
 Describe and determine signal properties including: power, energy,
periodic, time shifting, odd and even
 Understand the form of basic continuous and discrete time systems: first
and second order
 Describe and determine system properties including: linearity, time
invariance, superposition, stability, causality and system structures
Example Questions
1. Define the basic form of the discrete time impulse and step signals and
sketch the signals.
2. Calculate the power and energy of the following continuous and discrete
time signals: x(t) = e-4tu(t) and x[n] = (1/4)nu[n]
3. Are the following continuous and discrete time signal periodic, if so
determine their period (and fundamental frequency for the continuous time
signal): x(t) = jej5t, x[n] = ej7n.
4. Describe what system linearity means in terms of a system’s response to
individual signals. Is the following system linear: y[n] = 5*x[n] + 2?
5. Describe what system stability means. Are the following systems stable
y[n] = x[n], y(t) = t2*x(t), y[n] = 2y[n-1]+x[n]?
6. Describe what system causality means. Are the following discrete time
systems causal y[n] = x[n+1], y[n] = x[n-1]?
Many other similar questions are in SaS, O&W in 1.1-1.20, see the exercises
at the end of each lecture.
2. Convolution (Lectures 4-6)

Understand how a discrete time signal can be represented as the sum of
scaled and time shifted impulses (basis functions).

Understand why convolution can be used to analyse the response of linear
systems (superposition) using the impulse response of a system.
 Prove the convolution equation for discrete time, linear, time invariant
systems.
 Apply the discrete time convolution equation to signals and systems
involving step and power terms.
 Apply the continuous time convolution equation to signals and systems
involving step and exponential terms.
 Understand how the basic system properties of inverse, causality, stability
can be interpreted in terms of the system’s impulse response.
Example Questions
1. Describe how the sifting property can be used to represent a discrete time
signal in terms of impulse basis signals.
2. Describe how the superposition and sifting properties allows the response
of a discrete time, linear time invariant system to be determined by
y[n]  k  x[k ]h[n  k ] , where h[n] is the system’s impulse response.

3. Calculate, using convolution, the system’s response to the following input
signals and system with impulse response pairs
1. x[n] = (1/3)nu[n], h[n] = u[n]
2. x[n] = (1/3)nu[n], h[n] = u[n-2]
3. x[n] = u[n]-u[n-3], h[n] = u[n-2]
In all cases, sketch the input signal, the impulse response and the
system’s response.
4. Calculate, using convolution, the system’s response to the following input
signal and system impulse response pairs:
1. x(t) = e-4tu(t), h(t) = u(t)
2. x(t) = e-4tu(t), h(t) = u(t-2)
3. x(t) = u(t)-u(t-3), h(t) = u(t-2)
In all cases, sketch the input signal, the impulse response and the
system’s response.
5. Describe what system stability means in terms of the system’s impulse
response. Is the following discrete time system stable h[n] = n*u[n]?
6. Describe what system causality means in terms of the system’s impulse
response. Is the following system causal h(t) = e-t
Many other similar questions are in SaS, O&W in 2.1-2.11, 2.14, 2.15 see the
exercises at the end of each lecture.
3. Fourier Series and Transform (Lectures 7-9)

Understand how a continuous periodic signal can be represented as
complex linear combination of harmonic frequencies (basis functions) where
the base frequency is equal to the frequency of the original input signal.
 Calculate the Fourier series coefficients for periodic sinusoidal and step
signals.
 Plot the coefficients of a Fourier series as a function of frequency
(magnitude and phase).
 Understand why Fourier series only converge for signals with finite energy.
 Understand how the Fourier transform is defined for non-periodic signals.

Calculate the Fourier transforms for exponential, impulse, low pass filter
(rectangular step in the frequency domain).
 Calculate the inverse Fourier transform of simple frequency dependent
signals.
 Understand and apply the linearity and time shift properties for the Fourier
transform.
 Understand and apply the derivative of a Fourier transform.
 Solve ODEs using convolution in the frequency domain and designing low
pass filters using the inverse Fourier transform.
Example Questions
1. If x(t) is a periodic signal, describe the form of the basis functions that
occur in the Fourier series representation of the time domain signal.
2. Calculate the Fourier series (coefficients) of the following periodic signals
o x(t) = cos(t)
o x(t) = 2+cos((2/3)t)+4sin((5/3)t)
o x(t) = 1 (where 0<=t<1) and –1 (where 1<=t<2), which is period
2.
3. For each of the signals in question 2, sketch the time domain signals and
the magnitude/phase of the Fourier series coefficients
4. Does the Fourier transform of the following continuous time signals exist:
x(t) = e-5t, x(t) = et,
5. Calculate the Fourier transforms of the following continuous time signals:
o x(t) = e-5t
o x(t) = 1 (where |t|<1) and 0 otherwise
o x(t) = (t)
o x(t) = sin(t)
o x(t) = 2+cos((2/3)t)+4sin((5/3)t)
In each case, explicitly state whether the Fourier transforms exist, and
sketch the time domain signals and magnitude of the Fourier transforms
6. Using the inverse Fourier transform formula, calculate and sketch the time
domain signal corresponding to the frequency domain signal
o X(j) = 1 (where ||<5) and 0 otherwise
7. Show that the Fourier transform of dx/dt is jX(j)
8. Using convolution in the frequency domain, calculate the system’s
response when the input signal is x(t) = e-2tu(t) and the impulse response
is h(t) = e-tu(t).
Many other similar questions are in SaS, O&W in Chapters 3 and 4, see the
exercises at the end of each lecture.
4. Laplace Transform (Lectures 12-14)

Understand the relationship between the Fourier and the Laplace
transform, where the latter can be defined in terms of the Fourier series with
a convergence factor.
 Calculate the Laplace transform of exponential, sinusoidal and step
signals.
 Understand how the Region of Convergence defines which values of s for
which the Laplace transform converges.

Understand how, and be able, to factor a Laplace transform into the poles
and zeros.
 Understand how to invert a Laplace transform via partial fractions.
 Understand and apply the linearity and time shift properties for the Laplace
transform.
 Understand and apply the derivative of a Laplace transform.
 Solve ODEs using convolution in the s-domain using Laplace transforms.
Example Questions
1. Describe how the Laplace and the Fourier transform are related and what
are the conditions are they the same.
2. Calculate the Laplace transform and the associated regions of
convergence for the following signals:
o x(t) = cos(5t)u(t)
o x(t) = e-3tu(t)
o x(t) = u(t-2)
o x(t) = 2e-3(t-1)u(t-1) - e-3(t-2)u(t-2)
3. For each of the Laplace transforms in question (2) state the corresponding
poles and zeros of the Laplace transfer function and draw them in the
complex s-plane along with the corresponding regions of convergence.
4. Show that the Laplace transform of dx/dt is sX(s)
5. Using convolution in the s-domain, calculate the system’s response when
the input signals and impulse responses are:
o x(t) = e-2tu(t), h(t) = e-tu(t).
o x(t) = e-4tu(t), h(t) = u(t-2).
o x(t) = u(t), h(t) = e-tu(t)+e-2tu(t).
In each case, sketch the input signal, the impulse response and the
calculated system response.
Many other similar questions are in SaS, O&W in Chapter 9, see the
exercises at the end of each lecture.
5. Transfer Functions (Lectures 15-17)

Calculate the Laplace transform of the impulse response of a continuous
time, linear time invariant system, from the differential equation of a system.
 Understand the structure of low pass filters in both the frequency and time
domain and to be able to transform between the two.
 Plot the (Fourier) transfer function in terms of its magnitude and phase as
a function of frequency. Be able to calculate the magnitude and phase for
particular values of the frequency.
 Understand and plot Bode diagrams of the (Fourier) transfer function
which are log magnitude and phase plots against log frequency.
 Understand system stability and causality properties in terms of the
system’s transfer function and its ROC.
 Derive the transfer function for systems made from the combination of
simpler systems in series and feedback.
 Understand stability and causality properties for the combined transfer
function of simple systems.
Example Questions
1. Describe how the system transfer function may be obtained in two
different, but equivalent, ways if you are given either the system’s impulse
response or the differential equation that describes the system.
2. Calculate the (Laplace) transfer function of a system with an impulse
response
a. h(t) = e-a(t-1)u(t-1)
3. Calculate the (Laplace) transfer function of a system described by the
differential equation: dy(t)/dt + ay(t) = x(t-1)
4. Describe what high pass and low pass filters are in terms of their
frequency response properties.
You should describe how this is
determined by convolution in the frequency domain and describe/sketch
the filters’ impulse responses in the frequency domain.
5. Calculate the (Fourier) transfer function of the system with impulse
response h(t) = e-tu(t). Plot both its magnitude and phase as a function of
frequency, and explicitly evaluate their values at = 0,1 and 10.
6. Describe how the Bode plot of the transfer function describe in question
(5) differs from the normal magnitude/phase – frequency plots.
7. Describe how system stability can be deduced from the system’s transfer
function description. Is the following transfer function stable H(s) = 1/(s-5),
Re{s}>5?
8. Describe how system causality can be deduced from the system’s transfer
function description. Is the following transfer function causal H(s) = 1/(s1), Re{s}<1?
9. Derive the overall transfer function, when it is a structure composed of two
systems H1(s) and H2(s) in series, as described in Lecture 17. How do the
stability properties of the combined system relate to the individual
systems.
10. Derive the overall transfer function, when it is a structure composed of two
systems H1(s) and H2(s) arranged in feedback, as described in Lecture 17.
How are the stability properties of the combined system related to the
stability properties of the individual systems?
Many other similar questions are in SaS, O&W in Chapter 6 and 9, see the
exercises at the end of each lecture.