Fourier Series, Fourier Transform, DTFT, DFT, Relationship of Laplace and Fourier Transforms, Z-Transforms.

1. FREQUENCY DOMAIN ANALYSIS
OF LTI SYSTEM
CHAPTER-4

2. HISTORY OF FOURIERHISTORY OF FOURIER

3. Fourier Series and Fourier Transform
Jean Baptiste Joseph Fourier, French mathematician and
physicist
(03/21/1768 - 05/16/1830)

4. Fourier Series and Fourier Transform
Fourier Series
Any periodic function can be expressed as the sum of
sine(s) and /or cosine(s) of different frequencies, each
multiplied by a different coefficients
Fourier Transform
Any function that is not periodic can be expressed as the
integral of sines and /or cosines multiplied by a weighing
function

5. Fourier Series: Example

6. FOURIER SERIES
Trigonometric Fourier series:
( ) ( ) ( )∑∑
∞
=
∞
=
++=
1
0
1
00 sincos
n
n
n
n tnbtnaatx ωω
∫
+
=
0
)(
1
Tt
dttxa
∫
∫
∫
+
+
⋅=
⋅=
=
0
0
)sin()(
2
)cos()(
2
)(
1
0
0
0
0
0
0
Tt
t
n
Tt
t
n
t
dttntx
T
b
dttntx
T
a
dttx
T
a
ω
ω

7. FOURIER SERIES
Polar Fourier Series:
( ) ( )∑
∞
=
++=
1
00 cos
n
nn tnCCtx φω
[ ]+= nnn baC
2/122
Here, C0 is the Average value of x(t)
[ ]






−=
+=
n
n
n
nnn
a
b
baC
tanφ

8. FOURIER SERIES
Exponential Fourier Series:
( ) ∑
∞
−∞=
=
n
Tntj
neCtx 0/2π
∫
+
−
⋅=
0
0/2
0
)(
1
Tt
t
Tntj
n etx
T
C π

9. Dirichlet condition: Fourier series
It is the condition for the existence of Fourier series.
x(t) and its integrals are finite and single valued over period of
one cycle T0.
x(t) must have finite number of discontinuities in given interval.x(t) must have finite number of discontinuities in given interval.
x(t) should have only finite number of maxima and minima in
the given interval of time.
The x(t) is absolutely integrable : ∞<∫−
dttx
T
T
2/
2/
0
0
|)(|

10. Fourier Series: Example-1
Obtain quadrature Fourier Series for the rectangular
pulse train shown in figure:

11. Example-1:Hint
∫
+
=
0
)(
1
0
Tt
dttx
T
a
( ) ( ) ( )∑∑
∞
=
∞
=
++=
1
0
1
00 sincos
n
n
n
n tnbtnaatx ωω
∫
∫
∫
+
+
⋅=
⋅=
0
0
)sin()(
2
)cos()(
2
0
0
0
0
0
0
Tt
t
n
Tt
t
n
t
dttntx
T
b
dttntx
T
a
T
ω
ω

12. Fourier Series: Example-2
Obtain Fourier Series for the sawtooth waveform shown in
figure and plot its spectrum:
Hint: ( ) ∑
∞
−∞=
=
n
Tntj
neCtx 0/2π
∫
+
−
⋅=
0
0/2
0
)(
1
Tt
t
Tntj
n etx
T
C π

13. Properties of Fourier SeriesProperties of Fourier Series

14. Properties: Fourier series
1. Linearity
2.Time shifting
3.Time reversal
4.Time scaling
5. Frequency shifting5. Frequency shifting
6. Differentiation

15. 1. Linearity
It states that for two periodic signal x(t) and y(t) with T0,
Then,
( )
( ) n
SF
n
SF
bty
atx
→←
→←
.
.
Then,
( ) [ ] [ ]nnn
SF
BbAaCtBytAxtz +=→←+= .
)()(

16. 2. Time shifting
It states that for periodic signal x(t) withT0 period,
Then,
( ) n
SF
atx →← .
Then,
( ) natjkSF
ettx 00.
0
ω−
→←−

17. 3. Time reversal
It states that for periodic signal x(t) withT0 period,
Then,
( ) n
SF
atx →← .
Then,
( ) n
SF
atx −→←− .

18. 4. Time scaling
It states that for periodic signal x(t) withT0 period,
Then,
( ) n
SF
atx →← .
Then,
( ) ∑
∞
−∞=
⋅=
n
Ttnj
n eatx 0/2 απ
α

19. 5. Frequency shifting
It states that for periodic signal x(t) withT0 period,
Then,
( ) xn
SF
Ctx →← .
Then,
( )0
00 ./2
)( nnxyn
SFTtnj
CCetx −=→←⋅ π

20. 6. Differentiation
It states that for periodic signal x(t) withT0 period,
Then,
( ) n
SF
Ctx →← .
Then,
0
. 2
)(
T
nj
Ctx n
SF
dt
d π
⋅→←

21. Fourier TransformFourier Transform

22. Fourier Transform
ContinuousTime FourierTransform:
Or
Inverse FourierTransform:
( ) ( )∫
∞
∞−
−
⋅= dtetxX tjω
ω ( ) ( )∫
∞
∞−
−
⋅= dtetxfX ftj π2
Inverse FourierTransform:
Representation:
( ) ( )fXtx F
→←
( ) ( )∫
∞
∞−
⋅= dfefXtx ftj π
π
2
2
1
( )[ ] ( )[ ]txFtxF 1−

23. Fourier Transform
Amplitude and phase spectrum:
Where, |X(f)| is the amplitude spectrum of x(t) and
( ) ( ) )(
|| fj
efXfX θ
⋅=
Where, |X(f)| is the amplitude spectrum of x(t) and
ϴ(f) is the phase spectrum.

24. Properties of Fourier TransformProperties of Fourier Transform

25. Properties: Fourier series
1. Linearity
2.Time shifting
3.Time scaling
4. Frequency shifting
5. Differentiation in time domain5. Differentiation in time domain
6. Multiplication in time domain
7. Convolution in time domain

26. 1. Linearity
If Fourier transforms are given by
Then,
( ) ( )
( ) ( )fXtx
fXtx
F
F
22
11
→←
→←
Then,
Linear combination of inputs get transformed in linear
combination of their Fourier transforms.
( ) ( )[ ] ( ) ( )fXafXatxatxa F
22112211 +→←+

27. 2. Time shifting
If ,
Then,
( ) ( )fXtx F
→←
Then,
( ) ( )fXettx dftjF
d
π2−
→←−

28. 3. Time scaling
If ,
Then,
( ) ( )fXtx F
→←
Compression in Time domain Expansion in Frequency domain
Expansion in Time domain Compression in Frequency domain
( ) ( )α
α
α /
||
1
fXtx F
→←

29. 4. Frequency shifting
If,
Then,
( ) ( )fXtx F
→←
Then,
( )c
FTfj
ffXetx c
−→←⋅ π2
)(

30. 5. Differentiation in time domain
If,
Then,
( ) ( )fXtx F
→←
Then,
( )fXfjtx F
dt
d
⋅→← π2)(

31. 6. Multiplication in time domain
If,
Then,
( ) ( )
( ) ( )fXtx
fXtx
F
F
22
11
→←
→←
Then,
Multiplication in time domain Convolution in frequency domain
)()()()( 2121 fXfXtxtx F
∗→←⋅

32. 7. Convolution in time domain
If,
Then,
( ) ( )
( ) ( )fXtx
fXtx
F
F
22
11
→←
→←
Then,
Convolution in time domain Multiplication in frequency domain
)()()()( 2121 fXfXtxtx F
⋅→←∗

33. Relationship:
Laplace Transform-Fourier Transform
We know that hence,
LaplaceTransform,
( ) ( )∫
∞
+−
⋅= dtetxsX tj )( ωσ
ωσ js +=
( ) ( )∫
∞
∞−
−
⋅= dtetxsX st
If (i.e. ) then,
( ) ( )∫∞−
+−
⋅= dtetxsX tj )( ωσ
( ) ( ){ }∫
∞
∞−
−−
⋅⋅= dteetxsX tjt ωσ
ωjs = 0=σ ( ) ( )∫
∞
∞−
−
⋅= dtetxsX tjω

34. Introduction to DTFTIntroduction to DTFT
Discrete-Time FourierTransform

35. DTFT:
Discrete time FourierTransform:
DTFT is used for analysis of non-periodic discrete time signals.
Mathematically it is defined as:
( ) ( )∑
∞
−
⋅= nj
enxX ω
ω
Inverse- DTFT is given by:
( ) ( )∑−∞=
−
⋅=
n
nj
enxX ω
ω
( ) ( ) ωω
π
π
π
ω
deXnx nj
∫−
⋅=
2
1

36. The Z-Transform
Chapter-5

37. Z-Transform
Need of Z-Transform:
Ability to completely characterize signals & linear systems , in
most general ways possible
Stability of system can be determined easily
Mathematical calculations are reduced in Z-transform
By calculating Z-transform of given signal, DFT & FT can be
determined.
Solution of differential equations can be simplified

38. Z-Transform
Definition of Z-Transform:
Single Sided Z-Transform:
( )[ ] ( ) ( )∑
∞
=
−
⋅==
0n
n
znxzXtxZ
( ) ( )zXztx →←
Double Sided Z-Transform:
=0n
( )[ ] ( ) ( )∑
∞
−∞=
−
⋅==
n
n
znxzXtxZ

39. Z-Plane or Z-Domain
Im(Z)
Unit Circle
( )ωj
eX
Re(Z)
Unit Circle
ω
r=1
0
2π 0 2π
ω

40. ROC: Region Of Convergence
Region of Convergence (ROC) is defined as the set of all
‘z’ values for which X(z) is finite.
i.e. ( ) ( ) ∞<⋅= ∑
∞
−∞=
−
||||
n
n
znxzX
Significance of ROC:
ROC is going to decide whether the system is stable/unstable.
Also determine the type of sequence:
Causal/Non-Causal
Finite/Infinite

41. Example of ROC
)()( nuanx n
=
The z-transform is given by:
∑∑
∞
−
∞
−
== 1
)()()( nnn
azznuazX
×a
Region of convergence
∑∑ =−∞=
==
0
)()()(
nn
azznuazX
Which converges to:
azfor
az
z
az
zX >
−
=
−
= −1
1
1
)(
Clearly, X(z) has a zero at z = 0 and a pole at z = a.

42. Properties of ROCProperties of ROC
Region of Convergence

43. Properties of ROC
The ROC is a ring whose center is at origin.
ROC can not contain any pole.
If ROC of X(z) includes unit circle then and then only the
Fourier transform of discrete time sequence x(n)
converges.
The ROC must be a connected region.
For a finite duration sequence, the ROC is entire Z-plane
except z=0 & z=∞.

44. Application of Z-Transform in Image
Processing

45. Properties of Z-TransformProperties of Z-Transform

46. Properties of Z-Transform
1. Linearity
2.Time shifting
3. Scaling in Z-domain
4.Time Reversal
5. Differentiation in Z-domain5. Differentiation in Z-domain
6. Convolution of two sequences
7. InitialValueTheorem
8. FinalValue Theorem

47. 1. Linearity
If Z Transforms are given by
Then,
( ) ( )
( ) ( )zXnx
zXnx
z
z
22
11
→←
→←
Then,
ROC: Intersection of ROC of X1(z) and X2(z)
( ) ( )[ ] ( ) ( )zXazXanxanxa z
22112211 +→←+

48. 2. Time Shifting
If
Then,
( ) ( )zXnx z
→←
ROC: Same as ROC of X(z) except
z=0 if k>0
z=∞ if k<0
( ) ( )zXzknx kz −
→←−

49. 3. Scaling in Z-Domain
If
Then,
( ) ( )zXnx z
→←
Then,
ROC:
( ) 





→←⋅
a
z
Xnxa zn
21 |||||| razra <<

50. 4. Time Reversal
If
Then,
( ) ( )zXnx z
→←
Then,
ROC:
( ) ( )1−
→←− zXnx z
21
1
||
1
r
z
r
<<

51. 5. Differentiation in Z-domain
If
Then,
( ) ( )zXnx z
→←
Then,
ROC: Same as ROC of X(z)
( ) ( )zX
dz
d
znxn z
⋅−→←⋅

52. 6. Convolution of two sequences
If
Then,
( ) ( )
( ) ( )zXnx
zXnx
z
z
22
11
→←
→←
Then,
ROC: is atleast Intersection of ROC of X1(z) and X2(z)
)()()()( 2121 zXzXnxnx z
⋅→←∗

53. 7. Initial value theorem
If for causal sequence x(n)
Then,
( ) ( )zXnx z
→←
Then,
( ) ( )zX
z
x
∞→
=
lim
0

54. 8. Final value theorem
If for causal sequence x(n)
Then,
( ) ( )zXnx z
→←
Then,
( ) ( )zX
z
x
1
lim
→
=∞