1. Signals & systems Ch.3 Fourier Transform of Signals and LTI System
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2. Signals and systems in the Frequency domain
Fourier transform
Time [sec]
Frequency [sec-1, Hz]
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3. 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
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4. 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?
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5. 3.1 Introduction cont’ Fourier Series (FS) i) If ii) If
 Any periodic function(signal) with period T can be represented by weighted sum of orthonormal functions.
What is meaning of magnitude of Fk?
Think about equalizer in audio system.
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6. 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
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7. 3.3 Fourier Representations for Four Classes of signals
Periodic Signals: Fourier Series Representations
“Any periodic function can be represented by weighted sum of basic periodic function.”
Fourier said
(periodic) - (discrete)
(discrete) - (periodic)
Time Property
Periodic
Non-periodic
Continuous (t)
Fourier Series (FS)
Fourier Transform (CTFT)
Discrete [n]
Discrete-Time
Fourier Series (DTFS)
Fourier Transform (DTFT)
DTFS (3.4)
CTFS (3.5)
Periodic?
yes! With the period of N
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8. 3.3 Fourier Representations for Four Classes of signals cont’
3.3.2 Non-periodic Signals: Fourier-Transform Representations
(aperiodic) - (continuous)
(continuous) - (aperiodic)
Inverse continuous time Fourier Transform (CTFT)
(aperiodic) - (continuous)
(discrete) - (periodic)
Inverse discrete time Fourier Transform (DTFT)
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9. 3.4 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series
(periodic) - (discrete)
(discrete) - (periodic)
Inverse DFT (3.10)
DFT
Example 3.2 Determining DTFS Coefficients
Figure 3.5 Time-domain signal for Example 3.2.
Find the frequency-domain representation of the signal depicted in Fig. 3.5
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10. 3.4 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series cont’
Just inner product to orthonormal vectors in the 5 dimensional space!!
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11. 3.4 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series cont’
DC!!
Figure 3.6 Magnitude and phase of the DTFS coefficients for the signal in Fig. 3.5.
Recall “weighted sum of orthonormal vectors in the 5 dimensional space”
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12. 3.4 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series cont’
Example 3.3 Computation of DTFS Coefficients by Inspection
orthnormal
Determine the DTFS coefficients of , using the method of inspection.
Figure 3.8 Magnitude and phase of DTFS coefficients for Example 3.3.
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13. (impulse train) - (impulse train)
3.4 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series cont’
Example 3.4 DTFS Representation of an Impulse Train
Find the DTFS Coefficients of the N-periodic impulse train
Figure 3.9 A discrete-time impulse train with period N.
How about in other period?
Draw X[k] in the k-axis.
(impulse train) - (impulse train)
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14. 3.4 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series cont’
Example 3.5 The Inverse DTFS of periodic X[k]
Use Eq. (3.10) to determine the time-domain signal x[n] from the DTFS coefficients depicted in Fig. 3.10
Figure 3.10 Magnitude and phase of DTFS coefficients for Example 3.5.
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15. 3.4 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series cont’
Example 3.6 DTFS Representation of a Square Wave
Find the DTFS coefficients for the N-periodic square wave
Figure 3.11 Square wave for Example 3.6.
(3.15) 등비수열의 합
DC=mean=평균
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16. 3.4 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series cont’
DC, also
Then for all k.
M=0 (impulse train)
2M+1 = N?
(square) - (sinc)
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17. 3.4 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series cont’
Example 3.7 Building a Square Wave form DTFS Coefficients
symmetric
Evaluate one period of the Jth term in Eq. (3.18) and the 2J+1 term approximation
for J=1, 3, 5, 23, and 25.
(3.18)
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18. 3.4 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series cont’
Example 3.8 Numerical Analysis of the ECG
Evaluate the DTFS representations of the two electrocardiogram (ECG) waveforms depicted in Figs. 3.15(a) and (b).
Normal heartbeat.
Ventricular tachycardia.
Magnitude spectrum for the normal heartbeat.
Magnitude spectrum for ventricular tachycardia.
Figure 3.15 Electrocardiograms for two different heartbeats and the first 60 coefficients of their magnitude spectra.
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19. 3.5 Continuous-Time Periodic Signals: The Fourier Series
Any periodic function can be represented by weighted sum of basic periodic functions.
(periodic)  (discrete)
(continuous)  (aperiodic)
Inverse FT where (3.19) Recall “orthonormal”!!
FT (3.20)
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20. 3.5 Continuous-Time Periodic Signals: The Fourier Series cont’
Example 3.9 Direct Calculation of FS Coefficients
Determine the FS coefficients for the signal depicted in Fig
Solution :
X[0]=?
FT
Figure Time-domain signal for Example 3.9.
Figure 3.17 Magnitude and phase for Ex. 3.9.
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21. 3.5 Continuous-Time Periodic Signals: The Fourier Series cont’
Example 3.10 FS Coefficients for an Impulse Train. Determine the FS coefficients for the signal defined by
Solution :
(impulse train)  (impulse train)
Example 3.11 Calculation of FS Coefficients by Inspection
Determine the FS representation of the signal using the method of inspection.
Solution :
(3.21)
(3.22)
Figure Magnitude and phase for Ex
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22. 3.5 Continuous-Time Periodic Signals: The Fourier Series cont’
Example Inverse FS : Find the time-domain signal x(t) corresponding to the FS coefficients Assume that the fundamental period is
Solution :
등비수열의 합
Figure 3.20 FS coefficients for Problem 3.9.
Figure Square wave for Example 3.13.
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23. 3.5 Continuous-Time Periodic Signals: The Fourier Series cont’
Example 3.13 FS for a Square Wave Determine the FS representation of the square wave depicted in Fig
For k=0,
where
Figure 3.22 The FS coefficients, X[k], –50  k  50, for three square waves.
In Fig Ts/T = (a) 1/4 . (b) 1/16. (c) 1/64.
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24. 3.5 Continuous-Time Periodic Signals: The Fourier Series cont’
To/T=1/2? (DC)  (impulse)
To  0? (impulse train)  (impulse train)
T  ? (aperiodic)  (continuous)
(square)  (sinc)
(sinc)  (square)
Figure Sinc function sinc(u) = sin(u)/(u)
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25. 3.5 Continuous-Time Periodic Signals: The Fourier Series cont’
Example 3.14 Square-Wave Partial-Sum Approximation
Let the partial-sum approximation to the FS in Eq.(3.29), be given by
This approximation involves the exponential FS Coefficients with indices
Consider a square wave and Depict one period of the th term in this sum, and find for and 99.
Solution :
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26. (discrete)  (periodic)
3.6 Discrete-Time Non-periodic Signals : The Discrete-Time Fourier Transform
(discrete)  (periodic)
(3.31)
(3.32)
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27. 3.6 Discrete-Time Non-periodic Signals : The Discrete-Time Fourier Transform cont’
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
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28. 3.6 Discrete-Time Non-periodic Signals : The Discrete-Time Fourier Transform cont’
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.
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29. 3.6 Discrete-Time Non-periodic Signals : The Discrete-Time Fourier Transform cont’
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30. 3.6 Discrete-Time Non-periodic Signals : The Discrete-Time Fourier Transform cont’
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.
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31. 3.6 Discrete-Time Non-periodic Signals : The Discrete-Time Fourier Transform cont’
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)
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32. 3.6 Discrete-Time Non-periodic Signals : The Discrete-Time Fourier Transform cont’
Example 3.22 Two different moving-average systems
solution :
Figure 3.36
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33. 3.6 Discrete-Time Non-periodic Signals : The Discrete-Time Fourier Transform cont’
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.
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34. 3.7 Continuous-Time Non-periodic Signals : The Fourier Transform
(continuous aperiodic)  (continuous aperiodic)
Inverse CTFT (3.35)
CTFT (3.26)
Condition for existence of Fourier transform:
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35. 3.7 Continuous-Time Non-periodic Signals : The Fourier Transform cont’
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?
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36. 3.7 Continuous-Time Non-periodic Signals : The Fourier Transform cont’
Example 3.25 FT of a Rectangular Pulse
Find the FT of x(t).
Solution : (square)  (sinc)
Example (a) Rectangular pulse. (b) FT.
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37. 3.7 Continuous-Time Non-periodic Signals : The Fourier Transform cont’
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)
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38. 3.7 Continuous-Time Non-periodic Signals : The Fourier Transform cont’
Example 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)
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39. 3.7 Continuous-Time Non-periodic Signals : The Fourier Transform cont’
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.
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40. 3.7 Continuous-Time Non-periodic Signals : The Fourier Transform cont’
Solution :
Figure BPSK (a) rectangular pulse shapes
(b) raised-cosine pulse shapes.
the same power constraints
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41. 3.7 Continuous-Time Non-periodic Signals : The Fourier Transform cont’
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.
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42. 9.1 Linearity and Symmetry Properties of FT
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43. 9.1 Linearity and Symmetry Properties of FT cont’
3.9.1 Symmetry Properties : Real and Imaginary Signals
(3.37)
(real x(t)=x*(t))  (conjugate symmetric)
(3.38)
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44. 9.1 Linearity and Symmetry Properties of FT cont’
3.9.2 SYSMMEYRY PROPERTIES : EVEN/ODD SIGNALS
(even)  (real)
(odd)  (pure imaginary)  
For even x(t),
real
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45. (convolution)  (multiplication)
3.10 Convolution Property
CONVOLUTION OF NON-PERIODIC SIGNALS
(convolution)  (multiplication)
But given
change the order of integration 
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46. 3.10 Convolution Property cont’
Example 3.31 Convolution problem in the frequency domain
be the input to a system with impulse response
Find the output
Solution: 
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47. 3.10 Convolution Property cont’
Example 3.32 Find inverse FT’S by the convolution property
Use the convolution property to find , where 
Ex 3.32 (p. 261). (a) Rectangular z(t). (b)
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48. 3.10 Convolution Property cont’
FILTERING
Continuous time Discrete time
LPF
HPF
BPF
Figure (p. 263)
Frequency dependent gain (power spectrum)
kill or not (magnitude)
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49. 3.10 Convolution Property cont’
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 
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50. 3.10 Convolution Property cont’
EXAMPLE 3.35 Equalization 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 
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51. 3.10 Convolution Property cont’
Convolution of periodic signals : Cyclic convolution
Convolution in just one period Or better to derive in the frequency domain
EXAMPLE 3.36 Convolution of 2 periodic signals
Evaluate the periodic convolution of the sinusoidal signal  
Figure (p. 268) Square wave for Example 3.36.
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52. 3.10 Convolution Property cont’ 
Figure (p. 270) z(t) in Eq. (3.45) when J = 10.
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53. 3.11 Differentiation and Integration Properties
DIFFERENTIATION IN TIME
EXAMPLE 3.37 The differentiation property implies that
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54. 3.11 Differentiation and Integration Properties cont’
Example 3.38 Resonance in MEMS accelerometer
The MEMS accelerometer introduced in Section 1.10 is described by the differential equation 
Fig (p. 273)
n = 10,000 rad/s
Q = 2/5, Q = 1, and Q = 200.
Resonance in 10,000 rad/s
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55. 3.11 Differentiation and Integration Properties cont’
Example Use the differentiation property to find the FS of the triangular wave depicted in Fig. 3.59(a) 
Signals for (a) Triangular wave y(t). (b)
where
If we differentiate z(t) once more???
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56. 3.11 Differentiation and Integration Properties cont’
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).
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57. 3.11 Differentiation and Integration Properties cont’
Ex) Prove
Note where a=0
We know 
since linear 
Fig. a step fn. as the sum of a constant and a signum fn.
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58. 3.11 Differentiation and Integration Properties cont’
Common Differentiation and Integration Properties.
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59. 3.12 Time-and Frequency-Shift Properties
Time-Shift Property
Table 3.7 Time-Shift Properties of Fourier Representations
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60. 3.12 Time-and Frequency-Shift Properties cont’
Example)
Figure 3.62  
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61. 3.12 Time-and Frequency-Shift Properties cont’
Frequency-Shift Property
Recall
Table 3.8 Frequency-Shift Properties
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62. 3.12 Time-and Frequency-Shift Properties cont’
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  
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63. 3.12 Time-and Frequency-Shift Properties cont’
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   
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64. 3.13 Inverse FT by Using Partial-Fraction Expansions
Let then 
N roots,
 partial fraction 
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65. 3.13 Inverse FT by Using Partial-Fraction Expansions cont’
Inverse Discrete-Time Fourier Transform
where
Then
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66. 3.13 Inverse FT by Using Partial-Fraction Expansions cont’
Example 3.45 Inversion by Partial-Fraction Expansion
Solution:
Using the method of residues described in Appendix B, We obtain
And
Hence,
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67. 3.14 Multiplication (modulation) Property
Given and
Change of variable to obtain
(3.56)
Where
(3.57)
denotes periodic convolution.
Here, and are periodic.
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68. 3.14 Multiplication (modulation) Property cont’
Example) Windowing in the time domain
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.
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69. 3.14 Multiplication (modulation) Property cont’
Example 3.46 Truncating the sinc function
Sol) truncated by  
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70. 3.14 Multiplication (modulation) Property cont’
Example 3.47 Radar Range Measurement: RF Pulse Train
Solution)  
(3.59)
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71. 3.14 Multiplication (modulation) Property cont’
Figure 3.69 FS magnitude spectrum of FR pulse train for The result is depicted as a continuous curve, due to the difficulty of displaying 1000 steps
Table 3.9 Multiplication Properties
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72. 3.15 Scaling Properties 
(3.60)
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73. 3.15 Scaling Properties cont’
Example 3.48 SCALING A RECTANGULAR PULSE
Let the rectangular pulse 
Solution : 
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74. 3.15 Scaling Properties cont’
Example 3.49 Multiple FT Properties for x(t) when
Solution)
we define
Now we define
Finally, since
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75. 3.16 Parseval’s Relationships
Table 3.10 Parseval Relationships for the Four Fourier Representations
Representation
Parseval Relation FT FS
DTFT
DTFS
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76. 3.16 Parseval’s Relationships cont’
Example 3.50 Calculate the energy in a signal
Use the Parseval’s theorem
Solution)
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77. 3.17 Time –Bandwidth Product
Figure (p. 305) Rectangular pulse illustrating the inverse relationship between the time and frequency extent of a signal.
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78. 3.17 Time –Bandwidth Product cont’
Example 3.51 Bounding the Bandwidth of a Rectangular Pulse
Use the uncertainty principle to place a lower bound on the effective bandwidth of x(t).
Solution:
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79. 3.18.1 The Duality Property of the FT
Fig Duality of rectangular pulses and sinc functions
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80. 3.18.1 The Duality Property of the FT cont’
Example By using Duality, find the FT of
Solution)  
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81. 3.18.2 The Duality Property of the DTFS
&
The Duality Property of the DTFS
The Duality Property of the DTFT and FS   
Table 3.11 Duality Properties of Fourier Representations 
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82. 3.18.3 The Duality Property of the DTFT and FS cont’
Example 3.53 FS-DTFT Duality
Use the duality property and the results of Example 3.39 to determine the inverse DTFT of the triangular spectrum depicted in Fig 3.75(a).
Solution:
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