Pr

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N
SW ss
or
,A E
us . A
tra m
b
lia ik
a ira
ja
h
Part A: Signal Processing
 h
ja
ira
Chapter 1: Signal and Systems

a
lia ik
1.1 Signals

b
1.2 Sampling

tra m
1.3 Systems

us . A
1.4 Periodic Signals
1.5 Discrete-Time Sinusoidal Signals
,A E
1.6 Real Exponential Signals
or
1.7 Complex Exponential Signals
1.8 The Unit Impulse
SW ss

1.9 Simple Manipulations of Discrete-Time

U ofe

Signals
1.10 Problem Sheet A1 & MATLAB Exercises
Pr
N
Chapter 1: Signals and Systems

h
ja
1.0 Introduction

a ira
The terms signals and systems are given

lia ik
„
various interpretations. For example, a

b
tra m
system is an electric network consisting of

us . A
resistors, capacitors, inductors and energy
sources.
,A E
or

„ Signals are various voltages and currents in

SW ss

the network. The signals are thus functions of

U ofe

time and they are related by a set of

equations.
Pr
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 h
Example:

ja
ira
i(t) R

a
+ C +

lia ik
- i(t) vC(t)

b
-

tra m
us . A
,A E Figure: 1.0: An electric circuit

The objective of system analysis is to

or
„
SW ss

determine the behaviour of the system

U ofe

subjected to a specific input or excitation

Pr
N
 h
It is often convenient to represent a system

ja
„

ira
schematically by means of a box as shown in

a
Figure 1.1.

lia ik
b
tra m
us . A
,A EInput System Output
or

Figure: 1.1: General representation of a system.

SW ss
U ofe
Pr
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 h
1.1 Signals

ja
ira
„ There are two types of signals:

a
lia ik
(a) Continuous – time signals

b
tra m
(b) Discrete – time signals

us . A
In the case of a continuous-time signal, x(t), the
,A E
independent variable t is continuous and thus x(t) is
or
defined for all t (see Fig 1.2).
SW ss
U ofe

t – Continuous time -independent variable (-∞ < t < ∞)

Pr
N
 h
ja
„ On the other hand, discrete-time signals

ira
are defined only at discrete times and

a
consequently the independent variable

lia ik
takes on only a discrete set of values

b
tra m
(see Figure 1.2). A discrete- time signal

us . A
is thus a sequence of numbers.
,A E
or
„ n – discrete time - independent variable
SW ss

( n = … -2, -1, 0, 1, 2,…)

U ofe
Pr
N
Examples:

h
ja
1. A person’s body temperature is a

ira
continuous-time signal.

a
lia ik
b
2. The prices of stocks printed in the daily

tra m
newspapers are discrete-time signals.

us . A
,A E
3. Voltages & currents are usually represented
or
by continuous-time signals. They are
SW ss

represented also by discrete-time signals if

U ofe

they are specified only at a discrete set of

values of t.
Pr
N
 h
ja
a ira
lia ik
b
tra m
us . A
,A E
or
SW ss
U ofe
Pr

Figure 1.2: Above: An example of continuous-time signals. Below: An example of discrete-time signals.
N
1.2 Sampling

h
ja
„ A discrete-time signal is often formed by

ira
sampling a continuous -time signal x(t). If the

a
lia ik
samples are equidistant then

b
tra m
x[n] = x(t ) t = nT = x(nT )

us . A
(1.1)
,A E
or

Square brackets [ ] ⇒ Discrete time signals

SW ss

Round Brackets ( ) ⇒ Continuous signals

U ofe
Pr
N
 h
ja
Analogue Signal x(t) Digital signal

ira
x(nT ) = x[n]

a
1
fs =

lia ik
T

b
tra m
The constant T is the sampling interval or period and

us . A
,A E 1
the sampling frequency f s = Hz.
T
or
SW ss
U ofe
Pr
N
 h
ja
a ira
lia ik
b
tra m
us . A
,A E
or
SW ss

Figure 1.4: An example of acquiring discrete-time signals by

U ofe

sampling continuous-time signals.

Pr

x[n] = { 3.5, 4, 3.25, 2, 2.5, 3.0 }

N

n=-1 n=0 n=2 n=4

 h
ja
It is important to recognize that x[n] is

ira
„

only defined for integer values of n. It is

a
lia ik
not correct to think of x[n] as being

b
tra m
zero for n not an integer, say n=1.5.

us . A
x[n] is simply undefined for non-integer
,A E
or
values of n.
SW ss
U ofe
Pr
N
 h
Sampling Theorem:

ja
a ira
„ If the highest frequency contained in an

lia ik
analogue signal x(t) is fmax and the signal is

b
tra m
sampled at a rate fs ≥ 2 fmax then x(t) can

us . A
be exactly recovered from its sample values
,A E
using an interpolation function.
or
SW ss
U ofe
Pr
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 h
Example:

ja
ira
„ Audio CDs use a sampling rate, fs, of 44.1

a
lia ik
kHz for storage of the digital audio signal.

b
This sampling frequency is slightly more than

tra m
us . A
2×fmax [fmax = 20kHz], which is generally
accepted upper limit of human hearing and
,A E
perception of music sounds.
or
SW ss
U ofe
Pr
N
 Example:

h
ja
ira
⎧1 t ≥ 0

a
lia ik
u (t ) = ⎨ (1.2)

b
⎩0 t < 0

tra m
us . A
A continuous-time unit step function u(t)
„
,A E
or
is defined by [Fig 1.5].
SW ss
U ofe
Pr
N
 h
Note that the unit step is discontinuous

ja
„

ira
at t = 0. Its samples u[n] = u(t)|t=nT

a
form the discrete-time signal and

lia ik
b
defined by

tra m
us . A ⎧1 n ≥ 0
,A E
u[n] = ⎨ (1.3)
or

⎩0 n < 0
SW ss
U ofe
Pr
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 h
ja
a ira
lia ik
b
tra m
us . A
,A E
or
SW ss
U ofe
Pr

Figure 1.5: Top: Continuous-time unit step function. Bottom:

N

Discrete-time unit step function.

Exercise:

h
ja
Sketch the wave form: y[n] = u[n] − u[n − 1]

ira
„

a
lia ik
b
tra m
us . A
,A E
or
SW ss
U ofe
Pr
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 h
Exercise:

ja
Sketch the waveform for y (t ) = u (t + 1) − 2u (t ) + u (t − 1)

ira
„

a
lia ik
b
tra m
us . A
,A E
or
SW ss
U ofe
Pr
N
 h
1.3 Systems

ja
ira
„ A continuous-time system is one whose input x(t)

a
and output y(t) are continuous time functions related

lia ik
by a rule as shown in Fig 1.6(a).

b
tra m
y(t)
x(t)

us . A
Continuous y(t)
,A E x(t)
Time
t System t
or
SW ss

Fig 1.6 (a): General representation of continuous-time systems.

U ofe
Pr
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 h
A discrete system is one whose input x[n] and output

ja
„

y[n] are discrete time function related by a rule as

ira
shown in Fig 1.6(b).

a
lia ik
b
tra m
x[n] y[n]

us . A
x[n] y[n]
,A E Discrete
Time System
n n
or
SW ss

Fig 1.6 (b): General representation of discrete-time systems.

U ofe
Pr
N
 h
ja
„ An important mathematical

ira
distinction between continuous-time

a
lia ik
and discrete-time systems is the fact

b
tra m
that the former are characterized by

us . A
differential equations whereas the
,A E
latter are characterized by
or

difference equations.
SW ss
U ofe
Pr
N
Example:

h
ja
ira
„ The RC circuit shown in Figure 1.7 is a

a
continuous-time system

lia ik
b
output

tra m
i(t)

us . A
R +
,A E e(t) + C vC(t)
- i(t)
input -
or
SW ss

Figure 1.7: A diagram of RC circuit as an example

U ofe

of continuous-time systems.
Pr
N
 h
output

ja
i(t)

ira
R +
e(t) + C vC(t)

a
- i(t)

lia ik
input -

b
tra m
us . A
„ If we regard e(t) as the input signal
,A E
and vc(t) as the output signal, we
or
obtain using simple circuit analysis
SW ss

dvC (t ) 1 1
U ofe

+ vC (t ) = e(t ) (1.4)
dt RC RC
Pr
N
 h
ja
dvC (t ) 1 1
+ vC (t ) = e(t )

ira
(1.4)
dt RC RC

a
lia ik
b
From equation (1.4), a discrete -time system can

tra m
be developed as follows: If the sampling period T

us . A
is sufficiently small,
,A E
or

vC (nT ) − vC (nT − T )
SW ss

dvC (t )
= (1.5)
U ofe

dt t = nT T
Pr
N
 h
ja
vC(t) vC(nT)

ira
P
vC(nT)-vC(nT-T)

a
lia ik
T

b
tra m
us . A
nT-T nT t
,A E
Backward Euler approximation
or
[Assuming T is sufficiently small]
SW ss
U ofe

Figure 1.8: An approximation of discrete-time systems from the

continuous-time systems.
Pr
N
 By substituting equation (1.5) into (1.4)

h
„

ja
and replacing t by nT, we obtain:

ira
vC (nT ) − vC (nT − T ) 1

a
vC (nT ) = e(nT )
1
+

lia ik
T RC RC

b
tra m
The difference equation is:

us . A
vC [n] − vC [n − 1] 1 1
,A E
+ vC [n] = e[n]
T RC RC
or
SW ss

RC T difference
vC[n] = vC[n −1] + e[n] (1.6)
U ofe

equation
RC+T RC+T
Pr
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output previous output input

 h
ja
Summary:

a ira
Continuous-Time System

lia ik
b
tra m
Analogue input Differential Analogue output
Equations

us . A
,A E
or
SW ss

Digital input Difference Digital output

U ofe

Equations
Pr

Discrete-Time System
N
Example:

h
ja
ira
Analogue Signal Discrete-time signal

a
lia ik
1. x(t) = eat x[n] = eanT

b
t=nT

tra m
1 sampling
time fs = Hz frequency

us . A
Sample number Sampling T
,A E [0,1,2,3,…] Period (T)
or

2. x(t) = 10e-t – 5e- 0.5 t x[n] = 10e-nT – 5e- 0.5 nT

SW ss

t=nT
U ofe

sample number
Pr
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 x(t) = Acos(ωat) x[n] = A cos(ωanT)

h
3.
t=nT

ja
Analogue frequency

ira
in radians ωa = 2πfa
1
= A cos(2πf a ⋅ n ⋅ )

a
fs

lia ik
b
fa
= A cos(2π ⋅ n) = A cos(nθ )

tra m
fs

us . A
θ = digital frequency
,A E
fa
θ = 2π
or
fs
SW ss

θ=ωaT
U ofe

Exercise :
Pr

x(t) = A(1+m cos(ωmt))cos(ωct) x[n] =?

N
 h
1.4 Periodic Signals

ja
ira
„ An important class of signals is the periodic

a
signals. A periodic continuous-time signal x(t)

lia ik
has the property that there is a positive value

b
tra m
of P for which
x(t ) = x(t + P )

us . A
(1.7)
,A E
for all values of t.
or
„ In other words, a periodic signal has the
SW ss

property that is unchanged by a time shift of

U ofe

P. In this case we say x(t) is periodic with

Pr

period P.
N
Example :

h
period = P

ja
x(t)

a ira
lia ik
b
tra m
us . A
-P 0 P 2P t
Figure 1.9A: An example of periodic signals
,A E
or
„ Periodic signals are defined analogously in discrete
SW ss

time. A discrete-time signal x[n] is periodic with

U ofe

period N, where N is a positive integer, if for all

Pr

values of n. x[n] = x[n + N ] (1.8)

N
 Pr
Example :

U ofe
N
SW ss
or
,A E
us . A
tra m
b
lia ik
a ira
ja
h
 h
1.5 Discrete-Time Sinusoidal Signals

ja
ira
„ A continuous-time sinusoidal signal is given by
x(t ) = A sin (ωa t ) = A sin (2πf a t )

a
(1.9)

lia ik
b
fa = analogue frequency

tra m
A discrete - time sinusoidal signal may be expressed as

us . A
„

x[n] = x(t)|t=nT = x(nT)

,A E
fa
x[n] = A sin( nωaT ) = A sin( 2π
or
n) (1.10)
fs
SW ss

x[n] = A sin( nθ )
U ofe

1
Sampling frequency f s =
Pr

T fa
θ - Digital frequency θ = 2π = ω aT
N

(1.11)
fs
 h
„ A discrete-time signal is said to be periodic with a

ja
period length N, if N is the smallest integer for

ira
which

a
x[n + N ] = x[n]
∴

lia ik
A sin ((n + N )θ ) = A sin (nθ )

b
tra m
us . A
which can only be satisfied for all n if Nθ=2πk
,A E
(where k is an arbitrary integer)
or

∴
SW ss

2πk2πk
N= =
U ofe

θ 2π
fa
fs
Pr

see equation (1.11)

N
 h
ja
fs
N= k (1.12)

ira
fa

a
lia ik
8000

b
So if fa = 1000Hz & fs = 8000 Hz then N = = 8 samples

tra m
1000

us . A
,A E
An example of a sinusoidal sequence is shown in Fig 1.10.
or
SW ss
U ofe
Pr
N
 h
ja
a ira
lia ik
b
tra m
us . A
,A E
or
SW ss
U ofe

Figure 1.10: An example of sinusoidal sequences. The

⎛ 2πn ⎞
Pr

period, N, is 12 samples. x[n] = cos⎜ ⎟

⎝ 12 ⎠
N
Example:

h
ja
„ Determine the fundamental period of x[n],

ira
⎛ 2π π⎞
x [n] = 10 cos ⎜ n+ ⎟

a
⎝ 15 5⎠

lia ik
2π

b
digital frequency θ =

tra m
15
The fundamental period is therefore (see equation (1.12))

us . A
„

2πk
N=
,A E
θ
or
„ where k is the smallest integer for which N has an
SW ss

integer value. This is satisfied when k = 1.

U ofe

2π ⋅ 1
N= = 15 samples
2π
Pr
N

15
 h
Example:

ja
ira
„ The sinusoidal signal x[n] has fundamental

a
period N=10 samples. Determine the

lia ik
b
smallest θ for which x[n] is periodic:

tra m
us . A
2πk 2π
,A E θ= = k
N 10
or

Smallest value of θ is obtained when k = 1

SW ss

„
U ofe

2π π
∴ θ= = radians / cycle
Pr

10 5
N
 h
1.6 Real Exponential Signals

ja
ira
The continuous-time complex exponential

a
„

lia ik
signal is of the form

b
x(t ) = ce

tra m
at
(1.13)

us . A
where c and a are, in general complex
,A E
numbers. Depending upon the values of
or

these parameters, the complex exponential

SW ss

can exhibit several different characteristics.

U ofe
Pr
N
 x(t)

h
ja
Growing exponential a>0.

a ira
lia ik
b
c

tra m
t
x(t)

us . A
,A E Decaying exponential a<0
or
SW ss
U ofe

c
t
Figure 1.11: Characteristics of real exponential signals in terms of
Pr

time, t. Top: For a>0, the signal grows exponentially. Bottom: For
N

a<0, the signal decays exponentially.

1.7 Complex Exponential Signal [8]

h
ja
Consider a complex exponential, ceat where c is

ira
„
jθ
expressed in polar form, c = c e , and a in

a
lia ik
rectangular form, a = r + jω 0 .

b
Then

tra m
us . A
ce at =| c | e jθ e ( r + jω0 )t =| c | e rt ⋅ e j (ω0t +θ )
,A E =| c | e rt cos(ω0t + θ ) + j | c | e rt sin(ω0t + θ ) (1.14)
or
SW ss

„ Thus, for r = 0, the real & imaginary parts of a

U ofe

complex exponential are sinusoidal.

Pr
N
„ For r > 0 ⇒ Sinusoidal signals multiplied by a

h
ja
growing exponential

ira
For r < 0 ⇒ Sinusoidal signals multiplied by a

a
„

lia ik
decaying exponential [≅ damped sinusoids]

b
x(t)

tra m
x(t)
r<0

us . A
r>0
,A E
t
or
t
SW ss
U ofe
Pr

Growing sinusoidal signal Decaying sinusoidal signal

N

Figure 1.12: Characteristics of complex exponential signals.

 In discrete time, it is common practice to write a real

h
„

ja
exponential signal as x[n] = cαn (1.15)

a ira
„ If c and α are real and if |α|>1 the magnitude of the

lia ik
signal grows exponentially with n, while if |α|<1 we

b
tra m
have decaying exponential.

us . A
,A E
or
SW ss
U ofe
Pr
N
 h
ja
a ira
lia ik
b
tra m
us . A
,A E
or
SW ss
U ofe
Pr

Figure 1.13: Examples of discrete-time exponential signals.

N
 h
1.8 The Unit Impulse

ja
An important concept in the theory of linear systems

ira
„

is the continuous time unit impulse function. This

a
function, known also as the Dirac delta function is

lia ik
b
denoted by δ(t) and is represented graphically by a

tra m
vertical arrow.

us . A
,A E δ(t) Magnitude
1
1
or
SW ss
U ofe

0 t Frequency
Pr

Figure 1.14: Characteristic of the continuous-time impulse function

N

and the corresponding magnitude response in the frequency domain.

 h
„ The impulse function δ(t) is a signal of unit area

ja
vanishing everywhere except at the origin.

ira
∞

a
∫ δ ( t ) dt = 1 , δ ( t ) = 0 for t ≠ 0 (1.16)

lia ik
−∞

b
tra m
„ The impulse function δ(t) is the derivative of the step

us . A
function u(t).
du (t )
,A E
δ (t ) = (1.17)
dt
or
SW ss

u(t) du(t )
δ (t ) =
dt
U ofe

1 1
Pr

t t
N
 h
„ The discrete-time unit impulse function δ[n] is

ja
defined in a manner similar to its continuous time

ira
counterpart. We also refer δ[n] as the unit sample.

a
lia ik
b
tra m
us . A
⎧1 n = 0
δ [n] = ⎨
,A E (1.18)
⎩0 n ≠ 0
or
SW ss
U ofe
Pr
N

Figure 1.15: Characteristic of discrete-time impulse function.

1.9 Simple Manipulations of Discrete-

h
ja
Time Signals

ira
„ A signal x[n] may be shifted in time by

a
replacing the independent variable n by n-k

lia ik
b
where k is an integer.

tra m
us . A
„ If k>0 ⇒ the time shift results in a delay of
,A E
the signal by k samples [ie. shifting a signal
or

to the right]
SW ss
U ofe

„ If k<0 ⇒ the time shift results in an

Pr

advance of the signal by k samples.

N
 h
ja
a ira
lia ik
b
tra m
us . A
,A E
or
SW ss

Figure 1.16: Top left: Original signal, x[n]. Top right: x[n] is
U ofe

delayed by 2 samples. Bottom left: x[n] is advanced by 1 sample.

Advance: Shifting the signal to the left
Pr

Delay: Shifting the signal to the right

N
 h
Problem sheet A1:

ja
ira
Q1. A discrete – time signal x[n] is defined by

a
⎧1 0 ≤ n ≤ 9

lia ik
x[n] = ⎨

b
⎩0 otherwise

tra m
us . A
Using u[n], describe x[n] as the
,A E
or
superposition of two step functions.
SW ss
U ofe
Pr
N
 h
Q2. Sketch the following:

ja
ira
(a) x(t) = u(t-3) – u(t-5)

a
lia ik
(b) y[n] = u[n+3] – u[n-10]

b
(c) x(t) = e2tu(-t)

tra m
(d) y[n] = u[-n]

us . A
(e) x[n] = δ[n] + 2δ[n-1] -δ[n-3]
,A E
(f) h[n] = 2δ[n+1] + 2δ[n-1]
or
(g) h[n] = u[n], p[n] = h[-n]; q[n] = h[-1-n], r[n] = h[1-n]
SW ss

(h) x[n] = αn, α<1

U ofe

P[n] = αn u[n], q[n] = αnu[-n]

(i) x(t) = e-3t[u(t) – u(t-2)]
Pr
N
Q3. a) Consider a discrete-time sequence

h
ja
⎡ π π⎤

ira
x[n] = cos ⎢n + ⎥
⎣ 8 5⎦

a
lia ik
Determine the fundamental period of x[n].

b
(b) i) Consider the sinusoidal signal

tra m
x(t) = 10 sin(ωt) ω=2πfa

us . A
fa -analogue frequency and t- time,
,A E
fs -sampling frequency
or
SW ss

Write an equation for the discrete time signal x[n]

U ofe

ii) If fa = 200 Hz and fs = 8000 Hz, determine the

fundamental period of x[n].
Pr
N
 Summary of Part A: Chapter 1

h
ja
ira
„ At the end of this chapter, it is expected that
you should know:

a
lia ik
b
„ The difference between signals and systems

tra m
us . A
„ The sampling theorem, its limitations (e.g.
,A E
aliasing), and the sampling frequency (fs)
or

How to distinguish between continuous (analog)

SW ss

„
and discrete time (digital) signals
U ofe
Pr

„ How to distinguish between differential and

N

difference equations
 h
„ Continuous and discrete periodic signals

ja
and their definitions

a ira
„ The relationship between analog and digital

lia ik
b
2π f a

tra m
frequency θ =
fs

us . A
2π k fsk
The number of samples in a period: N = =
„
,A E
θ fa
θ = Digital frequency
or
SW ss
U ofe

„ Manipulation of discrete-time signals

Pr

„ The unit impulse and its properties

N