Signals & systems Ch.3 Fourier Transform of Signals and LTI System 5/30/2016

Signals and systems in the Frequency domain 5/30/2016KyungHee University2 Time [sec]Frequency [sec -1, Hz] Fourier transform

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Fundamental frequency of musical instruments MediaLab, Kyunghee University4

3.1 Introduction Orthogonal vector => orthonomal vector What is meaning of magnitude of H? 5/30/2016KyungHee University5 Any vector in the 2- dimensional space can be represented by weighted sum of 2 orthonomal vectors Fourier Transform(FT) Inverse FT

3.1 Introduction cont’ CDMA? 5/30/2016KyungHee University6 Orthogonal? Any vector in the 4- dimensional space can be represented by weighted sum of 4 orthonomal vectors Orthonormal function?

3.2 Complex Sinusoids and Frequency Response of LTI Systems 5/30/2016KyungHee University7 cf) impulse response How about for complex z? (3.1) How about for complex s? (3.3) Magnitude to kill or not? Phase  delay

Fourier transform 5/30/2016KyungHee University8 discrete time Continuous time Time domainfrequency domain Laplace transform z-transform (periodic)  -  (discrete) (discrete)  -  (periodic)

3.6 DTFT: Discrete-Time Fourier Transform 5/30/2016KyungHee University9 (discrete)  (periodic) (3.31) (3.32) (a-periodic)  (continuous)

5/30/2016KyungHee University10 Example 3.17 DTFT of an Exponential Sequence Find the DTFT of the sequence Solution : x[n] =  nu[n]. magnitude phase  = 0.5  = DTFT Example 3.17

5/30/2016KyungHee University11 Example 3.18 DTFT of a Rectangular Pulse Let Find the DTFT of Solution : (square)  (sinc) Figure 3.30 Example (a) Rectangular pulse in the time domain. (b) DTFT in the frequency domain. 3.6 DTFT Example 3.18

5/30/2016KyungHee University DTFT Example 3.18

5/30/2016KyungHee University13 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. 3.6 DTFT Example 3.19

5/30/2016KyungHee University14 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) 3.6 DTFT Example

5/30/2016KyungHee University15 Example 3.23 Multipath Channel : Frequency Response Solution : (a) a = 0.5ej2  /3. (b) a = 0.9ej2  / DTFT Example 3.23

3.7 CTFT 5/30/2016KyungHee University16 (continuous aperiodic)  (continuous aperiodic) Inverse CTFT (3.35) CTFT (3.26) Condition for existence of Fourier transform:

3.7 CTFT Example /30/2016KyungHee University17 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?

3.7 CTFT Example /30/2016KyungHee University18 Example 3.25 FT of a Rectangular Pulse Find the FT of x(t). Solution : (square)  (sinc) Example (a) Rectangular pulse. (b) FT.

3.7 CTFT Example /30/2016KyungHee University19 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)

3.7 CTFT Example /30/2016KyungHee University20 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)

3.7 CTFT Example /30/2016KyungHee University21 Example 3.29 Digital Communication Signals Rectangular (wideband) Separation between KBS and SBS. Narrow band Figure 3.44 Pulse shapes used in BPSK communications. (a) Rectangular pulse. (b) Raised cosine pulse.

3.7 CTFT Example /30/2016KyungHee University22 Figure 3.45 BPSK (a) rectangular pulse shapes (b) raised-cosine pulse shapes. Solution : the same power constraints

3.7 CTFT Example /30/2016KyungHee University23 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.

Fourier transform 5/30/2016KyungHee University24 Discrete time Continuous time Time domainfrequency domain

3.9.1 Linearity Property 5/30/2016KyungHee University25

3.9.1 Symmetry Properties Real and Imaginary Signals 5/30/2016KyungHee University26 (3.37) (real x(t)=x*(t))  (conjugate symmetric) (3.38)

3.9.2 Symmetry Properties of FT EVEN/ODD SIGNALS 5/30/2016KyungHee University27 (even)  (real) (odd)  (pure imaginary)  For even x(t), real

3.10 Convolution Property 5/30/2016KyungHee University28 (convolution)  (multiplication) But given change the order of integration 

3.10 Convolution Property Example /30/2016KyungHee University29 Example 3.31 Convolution problem in the frequency domain Input to a system with impulse response Find the output Solution: 

3.10 Convolution Property Example /30/2016KyungHee University30 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)

Filtering 5/30/2016KyungHee University31 Continuous time Discrete time(periodic with 2π LPF HPF BPF Figure 3.53 (p. 263) Frequency dependent gain (power spectrum) kill or not (magnitude)

3.10 Convolution Property Example /30/2016KyungHee University32 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 

3.10 Convolution Property Example /30/2016KyungHee University33 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 

3.11 Differentiation and Integration Properties 5/30/2016KyungHee University34 EXAMPLE 3.37 The differentiation property implies that

3.11 Differentiation and Integration Properties 5/30/2016KyungHee University35 예제 한 두개

DIFFERENTIATION IN FREQUENCY 5/30/2016KyungHee University36 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 Figure 3.60 (p. 275) Gaussian pulse g(t). and  Then (But, c=?)

Laplace transform and z transform 5/30/2016KyungHee University37

Integration 5/30/2016KyungHee University38  Ex) Prove Note where a=0 We know  Fig. a step fn. as the sum of a constant and a signum fn.  since linear

Differentiation and Integration Properties 5/30/2016KyungHee University39 Common Differentiation and Integration Properties.

Time-Shift Property 5/30/2016KyungHee University40 Table 3.7 Time-Shift Properties of Fourier Representations Fourier transform of time-shifted z(t) = x(t-t 0 ) Note that x(t-t 0 ) = x(t) * δ( -t 0 ) and

3.12 Time-and Frequency-Shift Properties 5/30/2016KyungHee University41 Example) Figure 3.62  

Frequency-Shift Property 5/30/2016KyungHee University42 Recall Table 3.8 Frequency-Shift Properties

Frequency-Shift Property 5/30/2016KyungHee University43 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  

3.12 Shift Properties Ex /30/2016KyungHee University44 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   

3.12 Shift Properties Ex /30/2016KyungHee University45 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

3.13 Inverse FT: Partial-Fraction Expansions Inverse FT by using 5/30/2016KyungHee University46  N roots,  partial fraction 

3.13 Inverse FT: Partial-Fraction Expansions Inverse FT by using 5/30/2016KyungHee University47 Let then  N roots,  partial fraction 

Inverse FT: Partial-Fraction Expansions 5/30/2016 KyungHee University 48

Inverse DTFT /30/2016KyungHee University49 where Then

Inverse FT: Partial-Fraction Expansions 5/30/2016 KyungHee University 50

Inverse DTFT by z-transform /30/2016KyungHee University51 where Then

3.13 Inverse FT Example /30/2016KyungHee University52 Example 3.45 Inversion by Partial-Fraction Expansion Solution: Using the method of residues described in Appendix B, We obtain And Hence,

3.13 Inverse FT Example /30/2016KyungHee University53 Example 3.45 Inversion by Partial-Fraction Expansion Solution: Using the method of residues described in Appendix B, We obtain And Hence,

3.14 Multiplication (modulation) Property 5/30/2016KyungHee University54 Given and Change of variable to obtain Where (3.56) (3.57) denotes periodic convolution. Here, and are -periodic.

Modulation property MediaLab, Kyunghee University55

3.14 Modulation Property Ex /30/2016 KyungHee University 56 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.

3.15 Scaling Properties 5/30/2016KyungHee University57  (3.60)

3.15 Scaling Properties Example /30/2016KyungHee University58 Example 3.48 SCALING A RECTANGULAR PULSE Let the rectangular pulse  Solution : 

3.15 Scaling Properties Example /30/2016KyungHee University59 Example 3.49 Multiple FT Properties for x(t) when Solution) we define Now we define Finally, since

3.15 Scaling Properties Example /30/2016KyungHee University60 Example 3.49 Multiple FT Properties for x(t) when Solution) we define Now we define Finally, since

3.16 Parseval’s Relationships 5/30/2016KyungHee University61 Represent ation Parseval Relation FT DTFT Table 3.10 Parseval Relationships for the Four Fourier Representations

3.16 Parseval’s Relationships Example /30/2016KyungHee University62 Example 3.50 Calculate the energy in a signal Use the Parseval’s theorem Solution)