Prepared by Professor Eliathamby Ambikairajah, Head of

School
Electrical Engineering and Telecommunications
University of New South Wales (UNSW), Australia

Modified and Delivered by Dr. T. Thiruvaran

Chapter 1
Chapter 1: Signals and Systems ......................................................... 2
1.1 Introduction............................................................................. 2
1.2 Signals..................................................................................... 3
1.2.1 Sampling ........................................................................... 4
1.2.2 Discrete-Time Sinusoidal Signals................................... 10
1.2.3 Discrete-time Exponential Signals ................................. 12
1.2.4 The Unit Impulse ............................................................ 13
1.2.5 Simple Manipulations of Discrete-Time Signals ........... 14
1.3 Systems ................................................................................. 15
1.4 Summary ............................................................................... 18
Chapter 1: Problem Sheet 1

Chapter 1 1
Chapter 1: Signals and Systems

1.1 Introduction
The terms ‘signals’ and ‘systems’ are given various
interpretations. For example, a system is an electric network
consisting of resistors, capacitors, inductors and energy sources.

Signals are various voltages and currents in the network. The

signals are thus functions of time and they are related by a set of
equations.

Example:
i(t) R
C +
+ vC(t)
- i(t) -

Figure 1.1: An electric circuit

The objective of system analysis is to determine the behaviour of

the system subjected to a specific input or excitation. It is often
convenient to represent a system schematically by means of a box
as shown in Figure 1.2.

Input System Output

Figure 1.2: General representation of a system.

Chapter 1 2
1.2 Signals
There are two types of signals:

(a) Continuous – time signals

(b) Discrete – time signals

In the case of a continuous-time signal, x(t), the independent

variable t is continuous and thus x(t) is defined for all t (see Figure
1.3).

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

On the other hand, discrete-time signals are defined only at

discrete times and consequently the independent variable takes on
only a discrete set of values (see Figure 1.3). A discrete-time
signal is thus a sequence of numbers.

n – discrete time - independent variable (n = … -2, -1, 0, 1, 2,…)

Examples:

 A person’s body temperature is a continuous-time signal.

 The prices of stocks printed in the daily newspapers are
discrete-time signals.
 Voltages and currents are usually represented by continuous-
time signals. They are also represented by discrete-time
signals if they are specified only at a discrete set of values of t.

Chapter 1 3
 3

2.5 Continuous-time signal

2

1.5

0.5

0
time (t)
3

2.5 Discrete-time signal

2

1.5

0.5

0
0 5 10 15 20
sample number (n)
Figure 1.3: An example of continuous-time and discrete-time signals

1.2.1 Sampling

A discrete-time signal is often formed by sampling a continuous

-time signal x(t). If the samples are equidistant then

xn  xt  t  nT  xnT  (1.1)

Square brackets [ ]  Discrete time signals

Round Brackets ( )  Continuous signals

, x(t) , x(n)
Amplitude

t 01 7 n

t - time n – sample number

Figure 1.4 An example of acquiring discrete-time signal by sampling

continuous-time signal

The constant T is the sampling interval or period and fs=1/T is the

sampling frequency.

Chapter 1 4
 4

Continuous-time T: sampling period

3 signal
x(t)
2

0
-T 0 T 2T 3T 4T

4 Discrete-time signal
x[n]
3

0
-1 0 1 2 3 4
sample number (n)

Figure 1.5: An example of acquiring discrete-time signals by sampling

continuous-time signals.

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

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

It is important to recognize that x[n] is only defined for integer

values of n. It is not correct to think of x[n] as being zero for n not
an integer, say n=1.5. x[n] is simply undefined for non-integer
values of n.

Sampling Theorem

If the highest frequency contained in an analogue signal x(t) is

fmax and the signal is sampled at a rate fs  2 fmax then x(t) can be
exactly recovered from its sample values using an interpolation
function.

For example, audio CDs use a sampling rate fs = 44.1kHz for

storage of the digital audio signal. This sampling frequency is
slightly more than 2fmax [fmax = 20kHz], which is generally
accepted upper limit of human hearing and perception of music
sounds.

Chapter 1 5
Exercise: What is the minimum sampling frequency, fs, required to avoid aliasing
when sampling a signal with the following spectrum?

Magnitude
a) 30 kHz
b) 60 kHz
c) 45 kHz
d) none of the
above
f (kHz)
-30 0 15 30 45

Unit step function

A continuous-time unit step function u(t) is shown below:

1 t  0
u(t )   (1.2)
0 t  0

Note that the unit step is discontinuous at t = 0. Its samples u[n]

= u(t)|t=nT form the discrete-time signal and defined by

1 n  0
u[ n]   (1.3)
0 n  0

Chapter 1 6
Example: Sketch the wave form: yn  un  un  1

Example: Sketch the waveform for yt   ut  1  2ut   0.5ut  2

u(t+1) 2u(t)

1 1

-3 -2 -1 0 1 2 3 4 t -3 -2 -1 0 1 2 3 4 t

0.5u(t-2) y(t)

1 1

-3 -2 -1 0 1 2 3 4 t -3 -2 -1 0 1 2 3 4 t

-1

Exercise:

Sketch the following:

1) x(t) = e-3t[u(t) – u(t-2)]

2) h[n] = u[n], p[n] = h[-n]; q[n] = h[-1-n], r[n] = h[1-n]

Chapter 1 7
 Example:

Continuous-time Signal Discrete-time signal

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

t=nT
time (t) 1 sampling
Sample number (n) Sampling fs  Hz frequency
[0,1,2,3,…] Period (T) T

2. x(t) = 10e-t – 5e- 0.5 t t=nT

x[n] = 10-nT – 5e- 0.5 nT
sample number (n)

3. x(t) = Acos(at) x[n] = A cos(anT)

t=nT

Analogue 1
 A cos(2 f a n )
Frequency (a) fs
in radians
a = 2fa fa
 A cos(2 n)
fs
x[n]  A cos(n )
fa
where   2   aT
fs
 : digital frequency(rad)
−𝜋 ≤ 𝜃 ≤ 𝜋

Chapter 1 8
Periodic Signals

An important class of signals is the periodic signals. A periodic

continuous-time signal x(t) has the property that there is a positive
value of P for which

xt   xt  P  (1.4)

for all values of t. In other words, a periodic signal has the

property that is unchanged by a time shift of P. In this case we
say x(t) is periodic with period P.

Example

x(t)
Period = P

-3P -2P -P P 2P 3P t

Figure 1.6: An example of a continuous-time periodic signal

Periodic signals are defined analogously in discrete time. A

discrete-time signal x[n] is periodic with period N, where N is a
positive integer, if for all values of n.

xn  xn  N  (1.5)

Chapter 1 9
Example:

x[n] with Period, N = 3 samples

1.2.2 Discrete-Time Sinusoidal Signals

A continuous-time sinusoidal signal is given by

xt   Asin at   Asin 2f at  (1.6)

fa : analogue frequency

A discrete - time sinusoidal signal may be expressed as

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

fa
x[n]  A sin(n aT )  A sin(2 n)
fs (1.7)
x[n]  A sin(n )
1
Sampling frequency: fs 
: T

fa
 - Digital frequency:   2  aT (1.8)
fs

Chapter 1 10
A discrete-time signal is said to be periodic with a period length
N, if N is the smallest integer for which

xn  N   xn
 A sinn  N    A sinn 

which can only be satisfied for all n if

N=2k (where k is an arbitrary integer)
2k 2k
N  
 f
2 a
fs see eq. (1.8)

fs
N k (1.9)
fa

For example, fa = 1000Hz and fs = 8000 Hz then N=fa / fs=8

samples (assume k is the smallest positive integer, i.e. k=1)

The following diagram depicts the cosine wave  2n  with

x[ n]  cos 
 12 
period N = 12 samples.

Figure 1.7 An example of a periodic sinusoidal sequence

Chapter 1 11
Example:
 2 
Determine the fundamental period of x [ n]  10 cos  n 
2𝜋  15 3
Digital frequency:  =
15
2𝜋𝑘
Recall (equation 1.9): N = ; k=1 is the smallest positive
𝜃
integer
2  1
2
 N= 15 = 15 samples

Example:
The sinusoidal signal x[n] has fundamental period N=10 samples.
Determine the smallest  for which x[n] is periodic:
2k 2
  k
N 10
Smallest value of  is obtained when k = 1
2 
   radians/ cycle
10 5

1.2.3 Discrete-time Exponential Signals

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

signal as

x[n] = cn (1.10)

If c and  are real and if ||>1 the magnitude of the signal grows
exponentially with n, while if ||<1 we have decaying
exponential.

Chapter 1 12
 x[n] = cn  >1, c > 0
x[n]

x[n] = cn 0< < 1, c > 0

x[n]

Figure 1.8: Examples of discrete-time exponential signals.

1.2.4 The Unit Impulse

An important concept in the theory of linear systems is the

continuous time unit impulse function. This function, known also
as the Dirac delta function is denoted by (t) and is represented
graphically by a vertical arrow.

(t) Magnitude
1 1

0 time (t) Frequency (f)

(a) (b)

Figure 1.9 Characteristics of impulse response function (a) in time domain and
(b) in the frequency domain

The impulse function (t) is the derivative of the step function

u(t).
du(t )
u(t)  (t ) 
du(t ) dt
 (t )  (1.11)
1 1
dt t
0 t 0
Chapter 1 13
The discrete-time unit impulse function [n] is defined in a
manner similar to its continuous time counterpart. We also refer
[n] as the unit sample.

1 n  0
 [ n]   (1.12)
0 n  0

1.2.5 Simple Manipulations of Discrete-Time Signals

A signal x[n] may be shifted in time by replacing the independent

variable n by n-k where k is an integer.

If k>0  the time shift results in a delay of the signal by k samples

[ie. shifting a signal to the right]

If k<0  the time shift results in an advance of the signal by k

samples. [ie. shifting a signal to the left]

Figure 1.10: (a,b) Original signal x[n], (c) x[n] is advanced by 1 sample,
(d) x[n] is delayed by 2 samples

Chapter 1 14
1.3 Systems
A continuous-time system is one whose input x(t) and output y(t)
are continuous time functions related by a rule as shown in
Figure 1.11.
x(t) y(t)
x(t) Continuous y(t)
Time
t System t

Figure 1.11: General representation of continuous-time systems.

A discrete system is one whose input x[n] and output y[n] are
discrete time function related by a rule as shown in Figure 1.12.

x[n] x[n] y[n] y[n]

Discrete
Time System
n n

Figure 1.12: General representation of discrete-time systems.

An important mathematical distinction between continuous-time

and discrete-time systems is the fact that the former are
characterized by differential equations whereas the latter are
characterized by difference equations.
Continuous-Time System

Analogue input Differential

Equations Analogue output

Difference
Digital input Digital output
Equations

Discrete-Time System
Chapter 1 15
Example: The RC circuit (known as a low-pass filter or an
integrator) shown in Figure 1.13 is a continuous-time system.

output
i(t)
R +
e(t) + C vC(t)
- i(t) -
input

Figure 1.13: A diagram of RC circuit as an example of continuous-time systems.

If we regard e(t) as the input signal and vc(t) as the output signal,
we obtain using simple circuit analysis

dvC (t ) 1 1
 vC (t )  e(t ) (1.13)
dt RC RC

From equation (1.13), a discrete -time system can be developed

as follows: If the sampling period T is sufficiently small,

dvC (t ) vC ( nT )  vC ( nT  T )

dt t  nT T (1.14)

vc(t)
P
vc(nT)
vc(nT)-vc(nT-T)
vc(nT-T)
T
Backward Euler
Approximation
*assume T is
sufficiently small

nT-T nT t
Figure 1.34: An approximation of discrete-time systems from the continuous-time systems.
Chapter 1 16
By substituting equation (1.14) into (1.13) and replacing t by nT,
we obtain:
vC nT   vC nT  T 
vC nT   enT 
1 1

T RC RC

The difference equation is:

vC [ n]  vC [ n  1] 1 1
 vC [ n]  e[ n]
T RC RC

e[ n] (1.10) difference
RC T
vC [ n]  vC [ n  1] 
RC  T RC  T equation

output previous output input

input
+ e(t)
-

Exercise:
i(t)

The RC circuit, known as a high-pass filter or a differentiator, is

i(t)

shown below. Derive a difference equation relating the output and

input of the circuit.
R

output
i(t)
-
vR(t)
+

C +
e(t) + vR(t)
R
output

- -
i(t)
input

Chapter 1 17
1.4 Summary
At the end of this chapter, it is expected that you should know:

 The difference between signals and systems

 The sampling theorem, its limitations (e.g. aliasing), and the
sampling frequency (fs)
 How to distinguish between continuous (analog) and discrete
time (digital) signals
 How to distinguish between differential and difference
equations
 Continuous and discrete periodic signals and their definitions
 The relationship between analog and digital frequency   2 fa
fs
2 k fsk
 The number of samples in a period: N  ,
 fa
 = Digital frequency
 Manipulation of discrete-time signals
 The unit impulse and its properties

Chapter 1 18