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

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

MediaLab , Kyunghee University 나이대별/성별 가청 음압 사람 말은 작아도 잘 듣는다. Voice (W60: 여성 60대) 2018-02-13 MediaLab , Kyunghee University

Fundamental frequency of musical instruments 2018-02-13 MediaLab , Kyunghee University

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

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

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

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

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

3.6 DTFT Example 3.17 Example 3.17 DTFT of an Exponential Sequence Find the DTFT of the sequence Solution :  = 0.5  = 0.9 x[n] = nu[n]. magnitude  = 0.5  = 0.9 phase 2/13/2018 KyungHee University

3.6 DTFT Example 3.18 Example 3.18 DTFT of a Rectangular Pulse Let Find the DTFT of Solution : (square)  (sinc) Figure 3.30 Example 3.18. (a) Rectangular pulse in the time domain. (b) DTFT in the frequency domain. 2/13/2018 KyungHee University

3.6 DTFT Example 3.18 2/13/2018 KyungHee University

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

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

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

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

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

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

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

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

3.7 CTFT Example 3.29 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. 2/13/2018 KyungHee University

3.7 CTFT Example 3.29 Solution : the same power constraints Figure 3.45 BPSK (a) rectangular pulse shapes (b) raised-cosine pulse shapes. the same power constraints 2/13/2018 KyungHee University

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

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

3.9.1 Linearity Property 2/13/2018 KyungHee University

(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

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

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

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

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

3.10.2 Filtering Continuous time Discrete time(periodic with 2π LPF HPF BPF Figure 3.53 (p. 263) Frequency dependent gain (power spectrum) kill or not (magnitude) 2/13/2018 KyungHee University

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

3.10 Convolution Property Example 3.35 EXAMPLE 3.35 Equalization(inverse) of multipath channel or Consider again the problem addressed in Example 2.13. 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

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

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

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. 3.60. and   Then (But, c=?) Figure 3.60 (p. 275) Gaussian pulse g(t). 2/13/2018 KyungHee University

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

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

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

3.12.1 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

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

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

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

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

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

3.13 Inverse FT: Partial-Fraction Expansions 3.13.1 Inverse FT by using  N roots,  partial fraction  2/13/2018 KyungHee University

3.13 Inverse FT: Partial-Fraction Expansions 3.13.1 Inverse FT by using Let then  N roots,  partial fraction  2/13/2018 KyungHee University

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

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

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

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

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 2011.5.4

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 2011.5.4

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

MediaLab , Kyunghee University Modulation property 2018-02-13 MediaLab , Kyunghee University

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

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

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

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

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

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

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