Signals

1
Signals Systems

• Book Digital Signal Processing Mulgrew et al
  Macmillan 1999
• System Physical or Mathematical Model
• Stimulate excite with input signal
• System modifies input to produce response, or
  output

H(s) H(z)
x(t)
y(t)
yn
xn
h(t)
Convolution
H(s) and H(z) are the system transfer functions,
continuous (Laplace) and discrete (Z)
2
Time Frequency Representations

• Fourier Transform
• x(t) X(?)
• Laplace Transform
• x(t) X(s)
• Z Transform
• x(t) X(z)

Transfer Functions Y(s) X(s)
Y(z) X(z)
3
Practical Significance

H(s) H(z)
x(t)
y(t)

yn
xn
h(t)
Convolution
Replaced by Multiplication
Transfer Functions Y(s) X(s)
H(s)
y(t) L-1H(s) . X(s)
yn Z-1H(z) . X(z)
H(z)
Y(z) X(z)
Thus, y(t) or yn can be analytically determined,
knowing the transforms of the input and the
system transfer function
4
Jean Baptiste Joseph Fourier Born 21 March 1768
in Auxerre, Bourgogne
Lagrange Laplace both taught Fourier
5
Signal Classification

• Continuous Discrete x(t) xn
• Periodic and Non-Periodic (or Aperiodic)
• periodic if x(t) x(t T) or xn
  xnN
• Energy Signals
• Power Signals

6
Examples

• Energy Signals
• Power Signals
• Consider the case of periodic signals

7
Fourier Overview

• Fourier Series
• Fourier Transform
• Discrete FT

8
Fourier Series(1)

• Periodic Signals (ie these are power signals)
  can be represented by a weighted sum of sine and
  cosines

9
Fourier Series (2)

• In complex form

(kgt0)
10
Fourier Series (3)

• So, any periodic function, x(t), can be
  represented by an infinite sum of sine and cosine
  terms

t
A0 represents the average (dc) term and by
inspection this is zero (prove this by applying
equation for Ak )
11
Fourier Series (4)

• To find Ak and Bk

V
t
Fix T phase
T
-V
Now only Bk terms are of interest
12
Fourier Series (5)

13
Fourier Series (6)

14
Fourier Series (7)

15
Fourier Series (8) - Odd Even

Here, x(t) is an odd function
t0
16
Fourier Series (9) - Odd Even

Here, x(t) is an even function
t0
17
Fourier Series (10)

cos k?0t
cos 1?0t
cos 0?0t
sin k?0t
1
sin 1?0t
A1
Ak
A0/2
B1
Bk
?
x(t)
18
Fourier Transform Pair

• Often derived from the Fourier Series

19
Fourier Series to Transform (1)
?
V
T
t
20
Fourier Series to Transform(2)
?
V
T
t
Sketch Xk for k -10,-9, 0,1,2, .. 10
for ?/T 0.5 ?/T 0.1 (NB Xk is a
discrete series!)
21
Fourier Transform Example
?
from Fourier series
V
t
Fourier transform definition

0
22
Fourier Series Example
?
V
T
t
It can be shown
23
Fourier Overview

• Fourier Series
• Fourier Transform
• Discrete FT

24
Discrete Fourier Transform DFT
DFT
IDFT
Differences 1/N normalizing factor
phase in DFT imaginary terms are -ve. Same
algorithm can be used for both DFT and
IDFT (with some post-operations)
25
DFT IDFT
Xk is complex (R j I) or cos( ) j
sin() xn is very often real - data from the
world then k 0, 1 . N/2 , since terms for
kN/2-m terms for N/2m
26
Vector/Matrix Interpretation
xn
Xk

Wkn exp(-j?2nk/N
27
REAL part cos() IMAG. part sin()
k
Anti-Symmetric about N/2
Symmetric about N/2
28
ExamplesREAL part cos() IMAG. part sin()
xn
29
Odd and Even Functions(1)
Even
f(t)f(-t)
Odd
f(t)-f(-t)
30
Odd and Even Functions(2)
Even
Odd
Even
Odd
31
Odd and Even Functions(3)
x(t) x(-t)
Even
x(t) x(-t)
Resultant generally non-zero Symmetry about N/2
in DFT ie do half the work only
Even
cos(n) cos(N-n)
f(t) f(-t)
32
Odd and Even Functions(4)
f(t) f(-t)
Even
f(t) f(-t)
Convolution resultant is always zero Anti-symmetr
y about N/2 in DFT
Odd
sin(n) -sin(N-n)
f(t) - f(-t)
33
Fourier/Laplace Transforms
Fourier
Laplace

• s ? j ?

Power signals such as unit step Lower limit 0
for real signals
34
Fourier/Laplace Transform of Unit Step
Fourier
Laplace
Solution replace u(t) with exp(- ?t).u(t)

• s ? j ?

Power signals such as unit step Lower limit 0
for real signals
35
(Dynamic) Systems

• System Physical or Mathematical Model
• Stimulate excite with input signal
• System modifies input to produce response, or
  output

36
Transfer Functions, Poles and Zeros (1)

• Poles and Zeros are roots of a transfer function
• drive the function to ?
• Zeros drive the function to zero

h(t)
x(t)
y(t)

H(s)
Eg
j?
?
s -b

• Zeros at s0 s -a
• Poles at s -b and ..
• (find the two other x and plot)

X
0
-a
H(s) and H(z) are the system transfer functions,
continuous (Laplace) and discrete (Z)
37
Transfer Functions, Poles and Zeros (2)

• Poles determine nature of response, oscillatory
  or not

1
j?
j?
s2
s1
?
?
Poles at
X
X
Key
4
j?

s2
X
3
?
s1
X
complex roots complex response
4
38
Frequency Response from TFs (1)
j?
EG1
?1
M (?1)
?
?
X
a
j?
EG2
?1
M (?1)
?
?

• Question
• Find phase at ?3db
• EG1 EG2

39
Frequency Response from TFs (2)
EG3
j?
?1
Mb(?1)
Ma(?1)
?
?a
?b
X
X
j?
a
EG4
?1
Mb(?1)
Ma (?1)
?
?
?b
X
a
40
Transfer Functions, Poles and Zeros (3)

• Poles determine nature of response, oscillatory
  or not

1
j?
j?
s2
s1
?
?
Poles at
X
X
Key
4
j?

s2
X
3
?
s1
X
complex roots complex response
4
41
Transfer Functions, Poles and Zeros (4)

• Poles determine nature of response, oscillatory
  or not

1
j?
j?
-b
-a
?
?
X
X
Poles at a -b
4
a b are complex
3
j?
j?

b
X
ab
?
?
X
X
complex roots give complex response
a
X
42
Transfer Functions, Poles and Zeros (5)

• Standard Form

2
1
j?
j?
Limit before oscillations
s2
s1
s1 s2
?
?
X
X
X
X
Poles at
Complex poles
j?
Limit of stability
X
s1 s2 are complex when ? lt 1
3

X
?
?
Complex pairs
complex poles give complex response
?n
X
X