1. Signals & systems Ch.3 Fourier Transform of Signals and LTI System
2/13/2018

2. Signals and systems in the Frequency domain
Fourier transform
Time [sec]
Frequency [sec-1, Hz]
2/13/2018
KyungHee University

3. MediaLab , Kyunghee University
나이대별/성별 가청 음압
사람 말은 작아도 잘 듣는다.
Voice
(W60: 여성 60대)
MediaLab , Kyunghee University

4. Fundamental frequency of musical instruments
MediaLab , Kyunghee University

5. Orthogonal vector => orthonomal vector
3.1 Introduction
Orthogonal vector => orthonomal vector
What is meaning of magnitude of H?
Any vector in the 2- dimensional space can be represented by weighted sum of 2 orthonomal vectors
Fourier Transform(FT)
Inverse FT
2/13/2018
KyungHee University

6. 3.1 Introduction cont’ CDMA? Orthogonal?
Any vector in the 4- dimensional space can be represented by weighted sum of 4 orthonomal vectors
Orthonormal function?
2/13/2018
KyungHee University

7. 3.2 Complex Sinusoids and Frequency Response of LTI Systems
cf) impulse response
How about for complex z? (3.1)
How about for complex s? (3.3)
Magnitude to kill or not? Phase  delay
2/13/2018
KyungHee University

8. Fourier transform Time domain frequency domain discrete time
Continuous time
z-transform
Laplace transform
(periodic) - (discrete)
(discrete) - (periodic)
2/13/2018
KyungHee University

9. 3.6 DTFT: Discrete-Time Fourier Transform
(discrete)  (periodic)
(a-periodic)  (continuous)
(3.31)
(3.32)
2/13/2018
KyungHee University

10. 3.6 DTFT Example 3.17 Example 3.17 DTFT of an Exponential Sequence
Find the DTFT of the sequence
Solution :
 =  = 0.9
x[n] = nu[n].
magnitude
 =  = 0.9
phase
2/13/2018
KyungHee University

11. 3.6 DTFT Example 3.18 Example 3.18 DTFT of a Rectangular Pulse
Let Find the DTFT of
Solution : (square)  (sinc)
Figure Example (a) Rectangular pulse in the time domain. (b) DTFT in the frequency domain.
2/13/2018
KyungHee University

12. 3.6 DTFT Example 3.18
2/13/2018
KyungHee University

13. 3.6 DTFT Example 3.19
Example 3.19 Inverse DTFT of a Rectangular Spectrum
Find the inverse DTFT of
Solution : (sinc)  (square)
Figure 3.31 (a) Rectangular pulse in the frequency domain. (b) Inverse DTFT in the time domain.
2/13/2018
KyungHee University

14. 3.6 DTFT Example 3.20-21 Example 3.20 DTFT of the Unit Impulse
Find the DTFT of
Solution : (impulse) - (DC)
Example 3.21 Find the inverse DTFT of a Unit Impulse Spectrum.
Solution : (impulse train)  (impulse train)
2/13/2018
KyungHee University

15. 3.6 DTFT Example 3.23
Example 3.23 Multipath Channel : Frequency Response
Solution :
(a) a = 0.5ej2/3. (b) a = 0.9ej2/3.
(a) a = 0.5ej2/3. (b) a = 0.9ej2/3.
2/13/2018
KyungHee University

16. 3.7 CTFT (continuous aperiodic)  (continuous aperiodic) CTFT (3.26)
Inverse CTFT (3.35)
Condition for existence of Fourier transform:
2/13/2018
KyungHee University

17. 3.7 CTFT Example 3.24 Example 3.24 FT of a Real Decaying Exponential
Find the FT of
Solution :
Therefore, FT not exists.
LPF or HPF? Cut-off from 3dB point?
2/13/2018
KyungHee University

18. 3.7 CTFT Example 3.25 Example 3.25 FT of a Rectangular Pulse
Find the FT of x(t).
Solution : (square)  (sinc)
Example (a) Rectangular pulse. (b) FT.
2/13/2018
KyungHee University

19. 3.7 CTFT Example 3.25
Example 3.26 Inverse FT of an Ideal Low Pass Filter!!
Fine the inverse FT of the rectangular spectrum depicted in Fig.3.42(a) and given by
Solution : (sinc) -- (square)
2/13/2018
KyungHee University

20. 3.7 CTFT Example 3.27-28 Example 3.27 FT of the Unit Impulse
Solution : (impulse) - (DC) 
Example 3.28 Inverse FT of an Impulse Spectrum
Find the inverse FT of
Solution : (DC)  (impulse)
2/13/2018
KyungHee University

21. 3.7 CTFT Example 3.29 Example 3.29 Digital Communication Signals
Rectangular (wideband)
Separation between KBS and SBS.
Narrow band
Figure Pulse shapes used in BPSK communications. (a) Rectangular pulse. (b) Raised cosine pulse.
2/13/2018
KyungHee University

22. 3.7 CTFT Example 3.29 Solution : the same power constraints
Figure BPSK (a) rectangular pulse shapes
(b) raised-cosine pulse shapes.
the same power constraints
2/13/2018
KyungHee University

23. 3.7 CTFT Example 3.29 rectangular pulse. One sinc Raised cosine pulse
3 sinc’s
The narrower main lobe, the narrower bandwidth.
But, the more error rate as shown in the time domain
Figure 3.47 sum of three frequency-shifted sinc functions.
2/13/2018
KyungHee University

24. Fourier transform Time domain frequency domain Discrete time
Continuous time
2/13/2018
KyungHee University

25. 3.9.1 Linearity Property
2/13/2018
KyungHee University

26. (real x(t)=x*(t))  (conjugate symmetric)
3.9.1 Symmetry Properties
Real and Imaginary Signals
(real x(t)=x*(t))  (conjugate symmetric)
(3.37)
(3.38)
2/13/2018
KyungHee University

27. 3.9.2 Symmetry Properties of FT
EVEN/ODD SIGNALS
(even)  (real)
(odd)  (pure imaginary)  
For even x(t),
real
2/13/2018
KyungHee University

28. (convolution)  (multiplication)
3.10 Convolution Property
(convolution)  (multiplication)
But given
change the order of integration 
2/13/2018
KyungHee University

29. 3.10 Convolution Property Example 3.31
Example 3.31 Convolution problem in the frequency domain
Input to a system with impulse response
Find the output
Solution: 
2/13/2018
KyungHee University

30. 3.10 Convolution Property Example 3.32
Example 3.32 Find inverse FT’S by the convolution property
Use the convolution property to find x(t), where 
Ex 3.32 (p. 261). (a) Rectangular z(t). (b)
2/13/2018
KyungHee University

31. 3.10.2 Filtering Continuous time Discrete time(periodic with 2π LPF
HPF
BPF
Figure (p. 263)
Frequency dependent gain (power spectrum)
kill or not (magnitude)
2/13/2018
KyungHee University

32. 3.10 Convolution Property Example 3.34
Example 3.34 Identifying h(t) from x(t) and y(t)
The output of an LTI system in response to an input is
Find frequency response and the impulse response of this system.
Solution:
But 
But note 
2/13/2018
KyungHee University

33. 3.10 Convolution Property Example 3.35
EXAMPLE 3.35 Equalization(inverse) of multipath channel or
Consider again the problem addressed in Example In this problem, a distorted received signal y[n] is expressed in terms of a transmitted signal x[n] as  
Then 
2/13/2018
KyungHee University

34. 3.11 Differentiation and Integration Properties
EXAMPLE 3.37 The differentiation property implies that
2/13/2018
KyungHee University

35. 3.11 Differentiation and Integration Properties
예제 한 두개
2/13/2018
KyungHee University

36. 3.11.2 DIFFERENTIATION IN FREQUENCY
Differentiate w.r.t. ω,
Then,
Example 3.40 FT of a Gaussian pulse
Use the differentiation-in-time and differentiation-in-frequency properties for the FT of the Gaussian pulse, defined by and depicted in Fig
and  
Then (But, c=?)
Figure (p. 275) Gaussian pulse g(t).
2/13/2018
KyungHee University

37. Laplace transform and z transform
2/13/2018
KyungHee University

38. 3.11.3 Integration  Ex) Prove Note where a=0 We know  since linear 
Fig. a step fn. as the sum of a constant and a signum fn.
2/13/2018
KyungHee University

39. Differentiation and Integration Properties
Common Differentiation and Integration Properties.
2/13/2018
KyungHee University

40. Time-Shift Property
Fourier transform of time-shifted z(t) = x(t-t0)
Note that x(t-t0) = x(t) * δ(-t0) and
Table 3.7 Time-Shift Properties of Fourier Representations
2/13/2018
KyungHee University

41. 3.12 Time-and Frequency-Shift Properties
Example)
Figure 3.62  
2/13/2018
KyungHee University

42. 3.12.2 Frequency-Shift Property
Recall
Table 3.8 Frequency-Shift Properties
2/13/2018
KyungHee University

43. 3.12.2 Frequency-Shift Property
Example 3.42 FT by Using the Frequency-Shift Property
Solution: We may express as the product of a complex sinusoid
and a rectangular pulse  
2/13/2018
KyungHee University

44. 3.12 Shift Properties Ex. 3.43
Example 3.43 Using Multiple Properties to Find an FT
Sol) Let and
Then we may write
By the convolution and differentiation properties
The transform pair   
2/13/2018
KyungHee University

45. 3.12 Shift Properties Ex. 3.43
Example 3.43 Using Multiple Properties to Find an FT
Sol) Let and
Then we may write
By the convolution and differentiation properties
The transform pair  s  
2/13/2018
KyungHee University

46. 3.13 Inverse FT: Partial-Fraction Expansions
Inverse FT by using 
N roots,
 partial fraction 
2/13/2018
KyungHee University

47. 3.13 Inverse FT: Partial-Fraction Expansions
Inverse FT by using
Let then 
N roots,
 partial fraction 
2/13/2018
KyungHee University

48. Inverse FT: Partial-Fraction Expansions
2/13/2018
KyungHee University

49. Inverse DTFT
3.13.2
where
Then
2/13/2018
KyungHee University

50. Inverse FT: Partial-Fraction Expansions
2/13/2018
KyungHee University

51. 3.13.2 Inverse DTFT by z-transform
where
Then
2/13/2018
KyungHee University

52. 3.13 Inverse FT Example 3.45
Example 3.45 Inversion by Partial-Fraction Expansion
Solution:
Using the method of residues described in Appendix B, We obtain
And
Hence,
2/13/2018
KyungHee University

53. 3.13 Inverse FT Example 3.45
Example 3.45 Inversion by Partial-Fraction Expansion
Solution:
Using the method of residues described in Appendix B, We obtain
And
Hence,
2/13/2018
KyungHee University

54. 3.14 Multiplication (modulation) Property
Given and
Change of variable to obtain
(3.56)
Where
(3.57)
denotes periodic convolution.
Here, and are periodic.
2/13/2018
KyungHee University

55. MediaLab , Kyunghee University
Modulation property
MediaLab , Kyunghee University

56. 3.14 Modulation Property Ex 3.46
Example 3.46 Truncating the sinc function
Sol) truncated by  
Figure 3.66 The effect of Truncating the impulse response of a discrete-time system. (a) Frequency response of ideal system. (b) for near zero. (c) for slightly greater than (d) Frequency response of system with truncated impulse response.
2/13/2018
KyungHee University

57. 3.15 Scaling Properties 
(3.60)
2/13/2018
KyungHee University

58. 3.15 Scaling Properties Example 3.48
Example 3.48 SCALING A RECTANGULAR PULSE
Let the rectangular pulse 
Solution : 
2/13/2018
KyungHee University

59. 3.15 Scaling Properties Example 3.49
Example 3.49 Multiple FT Properties for x(t) when
Solution)
we define
Now we define
Finally, since
2/13/2018
KyungHee University

60. 3.15 Scaling Properties Example 3.49
Example 3.49 Multiple FT Properties for x(t) when
Solution)
we define
Now we define
Finally, since
2/13/2018
KyungHee University

61. 3.16 Parseval’s Relationships
Table 3.10 Parseval Relationships for the Four Fourier Representations
Representation
Parseval Relation FT
DTFT
2/13/2018
KyungHee University

62. 3.16 Parseval’s Relationships Example 3.50
Example 3.50 Calculate the energy in a signal
Use the Parseval’s theorem
Solution)
2/13/2018
KyungHee University