Signals & systems Ch.3 Fourier Transform of Signals and LTI System 4/16/2017

Signals and systems in the Frequency domain Fourier transform Time [sec] Frequency [sec-1, Hz] 4/16/2017 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 4/16/2017 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? 4/16/2017 KyungHee University

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. 4/16/2017 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 4/16/2017 KyungHee University

3.3 Fourier Representations for Four Classes of signals 3.3.1 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 4/16/2017 KyungHee University

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) 4/16/2017 KyungHee University

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 4/16/2017 KyungHee University

3.4 Discrete-Time Periodic Signals: The Discrete-Time Fourier Series cont’ Just inner product to orthonormal vectors in the 5 dimensional space!! 4/16/2017 KyungHee University

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” 4/16/2017 KyungHee University

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. 4/16/2017 KyungHee University

(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) 4/16/2017 KyungHee University

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. 4/16/2017 KyungHee University

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 4/16/2017 KyungHee University

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) 4/16/2017 KyungHee University

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) 4/16/2017 KyungHee University

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. 4/16/2017 KyungHee University

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) 4/16/2017 KyungHee University

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. 3. 16. Solution : X[0]=? FT Figure 3.16 Time-domain signal for Example 3.9. Figure 3.17 Magnitude and phase for Ex. 3.9. 4/16/2017 KyungHee University

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 3.18 Magnitude and phase for Ex. 3.11. 4/16/2017 KyungHee University

3.5 Continuous-Time Periodic Signals: The Fourier Series cont’ Example 3.12 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 3.21 Square wave for Example 3.13. 4/16/2017 KyungHee University

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. 3.21. For k=0, where Figure 3.22 The FS coefficients, X[k], –50  k  50, for three square waves. In Fig. 3.21 Ts/T = (a) 1/4 . (b) 1/16. (c) 1/64. 4/16/2017 KyungHee University

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 3.23 Sinc function sinc(u) = sin(u)/(u) 4/16/2017 KyungHee University

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 : 4/16/2017 KyungHee University

(discrete)  (periodic) 3.6 Discrete-Time Non-periodic Signals : The Discrete-Time Fourier Transform (discrete)  (periodic) (3.31) (3.32) 4/16/2017 KyungHee University

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 :  = 0.5  = 0.9 x[n] = nu[n]. magnitude  = 0.5  = 0.9 phase 4/16/2017 KyungHee University

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 3.30 Example 3.18. (a) Rectangular pulse in the time domain. (b) DTFT in the frequency domain. 4/16/2017 KyungHee University

3.6 Discrete-Time Non-periodic Signals : The Discrete-Time Fourier Transform cont’ 4/16/2017 KyungHee University

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. 4/16/2017 KyungHee University

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) 4/16/2017 KyungHee University

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 4/16/2017 KyungHee University

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. 4/16/2017 KyungHee University

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: 4/16/2017 KyungHee University

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? 4/16/2017 KyungHee University

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 3.25. (a) Rectangular pulse. (b) FT. 4/16/2017 KyungHee University

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) 4/16/2017 KyungHee University

3.7 Continuous-Time Non-periodic Signals : The Fourier Transform cont’ 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) 4/16/2017 KyungHee University

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 3.44 Pulse shapes used in BPSK communications. (a) Rectangular pulse. (b) Raised cosine pulse. 4/16/2017 KyungHee University

3.7 Continuous-Time Non-periodic Signals : The Fourier Transform cont’ Solution : Figure 3.45 BPSK (a) rectangular pulse shapes (b) raised-cosine pulse shapes. the same power constraints 4/16/2017 KyungHee University

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. 4/16/2017 KyungHee University

9.1 Linearity and Symmetry Properties of FT 4/16/2017 KyungHee University

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) 4/16/2017 KyungHee University

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 4/16/2017 KyungHee University

(convolution)  (multiplication) 3.10 Convolution Property 3.10.1 CONVOLUTION OF NON-PERIODIC SIGNALS (convolution)  (multiplication) But given change the order of integration  4/16/2017 KyungHee University

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:  4/16/2017 KyungHee University

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) 4/16/2017 KyungHee University

3.10 Convolution Property cont’ 3.10.2 FILTERING Continuous time Discrete time LPF HPF BPF Figure 3.53 (p. 263) Frequency dependent gain (power spectrum) kill or not (magnitude) 4/16/2017 KyungHee University

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  4/16/2017 KyungHee University

3.10 Convolution Property cont’ EXAMPLE 3.35 Equalization 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  4/16/2017 KyungHee University

3.10 Convolution Property cont’ 3.10.3 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 3.56 (p. 268) Square wave for Example 3.36. 4/16/2017 KyungHee University

3.10 Convolution Property cont’   Figure 3.57 (p. 270) z(t) in Eq. (3.45) when J = 10. 4/16/2017 KyungHee University

3.11 Differentiation and Integration Properties 3.11.1 DIFFERENTIATION IN TIME EXAMPLE 3.37 The differentiation property implies that 4/16/2017 KyungHee University

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 3.58 (p. 273) n = 10,000 rad/s Q = 2/5, Q = 1, and Q = 200. Resonance in 10,000 rad/s 4/16/2017 KyungHee University

3.11 Differentiation and Integration Properties cont’ Example 3.39 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??? 4/16/2017 KyungHee University

3.11 Differentiation and Integration Properties cont’ 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). 4/16/2017 KyungHee University

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. 4/16/2017 KyungHee University

3.11 Differentiation and Integration Properties cont’ Common Differentiation and Integration Properties. 4/16/2017 KyungHee University

3.12 Time-and Frequency-Shift Properties 3.12.1 Time-Shift Property Table 3.7 Time-Shift Properties of Fourier Representations 4/16/2017 KyungHee University

3.12 Time-and Frequency-Shift Properties cont’ Example) Figure 3.62   4/16/2017 KyungHee University

3.12 Time-and Frequency-Shift Properties cont’ 3.12.2 Frequency-Shift Property Recall Table 3.8 Frequency-Shift Properties 4/16/2017 KyungHee University

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   4/16/2017 KyungHee University

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    4/16/2017 KyungHee University

3.13 Inverse FT by Using Partial-Fraction Expansions Let then  N roots,  partial fraction  4/16/2017 KyungHee University

3.13 Inverse FT by Using Partial-Fraction Expansions cont’ 3.13.2 Inverse Discrete-Time Fourier Transform where Then 4/16/2017 KyungHee University

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, 4/16/2017 KyungHee University

3.14 Multiplication (modulation) Property Given and Change of variable to obtain (3.56) Where (3.57) denotes periodic convolution. Here, and are -periodic. 4/16/2017 KyungHee University

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. 4/16/2017 KyungHee University

3.14 Multiplication (modulation) Property cont’ Example 3.46 Truncating the sinc function Sol) truncated by   4/16/2017 KyungHee University

3.14 Multiplication (modulation) Property cont’ Example 3.47 Radar Range Measurement: RF Pulse Train Solution)   (3.59) 4/16/2017 KyungHee University

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 4/16/2017 KyungHee University

3.15 Scaling Properties  (3.60) 4/16/2017 KyungHee University

3.15 Scaling Properties cont’ Example 3.48 SCALING A RECTANGULAR PULSE Let the rectangular pulse  Solution :  4/16/2017 KyungHee University

3.15 Scaling Properties cont’ Example 3.49 Multiple FT Properties for x(t) when Solution) we define Now we define Finally, since 4/16/2017 KyungHee University

3.16 Parseval’s Relationships Table 3.10 Parseval Relationships for the Four Fourier Representations Representation Parseval Relation FT FS DTFT DTFS 4/16/2017 KyungHee University

3.16 Parseval’s Relationships cont’ Example 3.50 Calculate the energy in a signal Use the Parseval’s theorem Solution) 4/16/2017 KyungHee University

3.17 Time –Bandwidth Product Figure 3.72 (p. 305) Rectangular pulse illustrating the inverse relationship between the time and frequency extent of a signal. 4/16/2017 KyungHee University

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: 4/16/2017 KyungHee University

3.18.1 The Duality Property of the FT Fig. 3.73 Duality of rectangular pulses and sinc functions 4/16/2017 KyungHee University

3.18.1 The Duality Property of the FT cont’ Example 3.52 By using Duality, find the FT of Solution)   4/16/2017 KyungHee University

3.18.2 The Duality Property of the DTFS 3.18.2 & 3.18.3 3.18.2 The Duality Property of the DTFS 3.18.3 The Duality Property of the DTFT and FS    Table 3.11 Duality Properties of Fourier Representations  4/16/2017 KyungHee University

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: 4/16/2017 KyungHee University