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1 Lecture on Communication Theory Chapter 2. Representation of signals and systems 2.1 Introduction Deterministic signals : A class of signals where waveforms are defined exactly as function of time. 2.2 F.T. 1) Let g(t) = non-periodic deterministic signal FT of g(t) inverse F.T. 2) To exist FT of g(t), sufficient condition (not necessary) Dirichlet’s conditions g(t) is single-valued, with a finite number of maxima and minima in any finite time interval g(t) has a finite # of discontinuities in any finite time interval g(t) is absolutely integrable, that is, 3) Physical realizability existence of F.T. 4) All energy signal Fourier transformable g(t) = G(f) = where f = frequency

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2 Lecture on Communication Theory 2.2 Fourier Transform 1. Notation t : time [sec] f : frequency [Hertz] w= 2 f : angular frequency [radians/sec] 식 G(f) = F [g(t)] 식 g(t) = F -1 [G(f)] F, F -1 : linear operator 2. Continuous Spectrum 1) Continuous Spectrum G(f) = |G(f)| exp[j (f)] |G(f)| : Continuous amplitude spectrum of g(t) (f) : Continuous phase spectrum of g(t) 2) For real-valued g(t) G(f) = G * (f) = |G(-f)| = | G(f)| (-f) = - (f) Complex conjugation = conjugate symmetric Real : y 축 대칭, symmetric Imaginary : 원점 대칭, anti-symmetric g(t) G(f)

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3 Lecture on Communication Theory Ex1) Rectangular pulse Def) rect(t) = 1 -1/2 < t < 1/2 0 |t| > 1/2 g(t) = A rect (t/T) Def) sinc function G(f) = AT sinc (fT) A rect(t/T) AT sinc (fT) 3) Real Symmetric Real Symmetric Ex2) exp(-at) u(t) exp(at) u(-t) T/2 -T/2 A 0 AT 1/T-1/T FT

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4 Lecture on Communication Theory 2.3 Properties of the Fourier Transform 1. Linearity (Superposition) Let g 1 (t) G 1 (f) and g 2 (t) G 2 (f). Then for all constants C 1 and C 2, we have C 1 g 1 (t) + C 2 g 2 (t) C 1 G 1 (f)+ C 2 G 2 (f) 2. Time Scaling Let g(t) G(f). Then g(at) 1/|a| G(f/a) when a=-1, g(-t) G(-f) 3. Duality If g(t) G(f), then G(t) g(-f) 4. Time shifting If g(t) G(f), then g(t-t 0 ) G(f)exp(-j2 ft 0 ) 5. Frequency shifting If g(t) G(f), then exp(j2 f c t)g(t) G(f-f c ) where f c is a real constant modulation theorem

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5 Lecture on Communication Theory Ex5) RF pulse g(t) =a rect(t/T)cos(2 f c t) G(f) = AT/2 {sinc[T(f-f c )] + sinc[T(f+f c )]} 6. Area under g(t) If g(t) G(f), then 7. Area under G(f) If g(t) G(f), then 8. Differentiation in the Time Domain Let g(t) G(f), and assume that the 1st derivative of g(t) is Fourier transformable. Then n-th generalization |G(f)| T -f c fcfc g(t) Ex6) Gaussian pulse g(t) = exp(-pt 2 ) exp(-pf 2 )

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6 Lecture on Communication Theory 9. Integration in the Time Domain Let g(t) G(f), then provided that G(0)=0, we have 10. Conjugate Functions If g(t) G(f), then for a complex-valued time function g(t), we have g * (t) G * (-f), when the asterisk denotes the complex conjugate operation g * (-t) G * (f) 11. Multiplication in the Time Domain (Multiplication Theorem ) Let g 1 (t) G 1 (f) and g 2 (t) G 2 (f), Then g 1 (t)g 2 (t) 12. Convolution in the Time Domain (Convolution Theorem) Let g 1 (t) G 1 (f) and g 2 (t) G 2 (f), Then g 1 (t) g 2 (t) G 1 (f)G 2 (f)

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7 Lecture on Communication Theory 2.4 Rayleigh’s Energy Theorem 1. Rayleigh’s Energy Theorem E = linear transform Ex9) E =

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8 Lecture on Communication Theory 2.5 The inverse relationship btw time and frequency 1. Time 과 frequency 관계 1) linear transform of FT 2) Time Frequency wide low narrow high cannot specify arbitrary function of time and frequency. 3) Strictly limited in time infinite in freq. infinite in time strictly band-limited in freq. cannot be strictly limited in both time and freq. 2. Bandwidth 1) def) extent of significant spectral content of the signal for signal positive frequencies 2) strictly band-limited 경우가 아닐 경우의 definitions BW = Main lobe bounded by well-defined nulls 0 1/T-1/T 0 f c + 1/T fcfc modulation Base-bandPass-band 2 배 차이 BW = 1/T BW = 2/T

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9 Lecture on Communication Theory 3 dB Bandwidth : of its peak value rms(root mean square) BW ( 장점 ) mathematical evaluation ( 단점 ) not easily measurable in lab 3. Time-Bandwidth Product 1) for base-band sync function = (duration T) (BW of main lobe = 1/T) =1 2) T rms W rms 1/4 “ = “ when Gaussian pulse 2.6 Dirac Delta Function 1. Definition (t) = 0, t 0 (t) is even function of t 2. Shifting property W rms

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10 Lecture on Communication Theory 3. Replication property g(t) (t) = g(t) 4. Fourier Transform F [ (t)] = 1 5. Applications of the Delta Function 1) dc signal 1 (f) 2) Complex Exponential Function exp(j2 f c t) (f-f c ) 3) Sinusoidal Functions cos (2 f c t) = 1/2 [exp (j2 f c t) + exp (-j2 f c t)] 1/2 [ (f-f c ) + (f+f c )] Sin(2 f c t) 1/2j [ (f-f c ) - (f+f c )] g(t) t G(f) f 1.0 f f

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11 Lecture on Communication Theory 4) signum Function sgn[t] = +1, t > 0 0, t = 0 -1, t < 0 exp(-|a|t)sgn(t) F[sgn(t)] = 5) Unit step function u(t) = 1, t > 0 1/2, t = 0 0, t < 0 u(t) = 1/2 [sgn(t) + 1] u(t) g(t) 0 + 1.0 - 1.0 t |G(f)| 0 f f 1/2 g(t) t

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12 Lecture on Communication Theory 6) Integration in the Time Domain (Revisited) Let

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13 Lecture on Communication Theory 2.7 Fourier Transforms of periodic signals Let a periodic signal be g To (t) with period T 0 complex exponential Fourier series where f 0 = 1/T 0 ; fundamental freq. Generating function g(t) g(t) = g To (t), - T 0 /2 t T 0 /2 0, elsewhere from where G(f) g(t) (Observations) periodicity in time domain discrete spectrum defined at nf 0 Use Poisson’s sum formula 1 2 3 2 1 3

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14 Lecture on Communication Theory ex11)

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15 Lecture on Communication Theory g(t) sampling 1 G(f)* G(f) sampling 2 G(f)* 0 T2T t t t f f f f f 0 0 0 0 0 0 0 T T 2/T T/2 T/8 2.1, 2.4, 2.8, 2.17, 2.18

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16 Lecture on Communication Theory 2.8 Transmission of signals through linear system Def) linear system : a system which holds the principle of superposition 1. Time Response 1) Impulse Response : the response of system to (t) 2) Convolution y(t) = excitation time response time t system - memory time t- ; a weighted integral over the past history of the input signal, weighted according to the impulse response of the system. h(t) x 1 (t) y 1 (t) h(t) x 2 (t) y 2 (t) h(t) c 1 x 1 (t) + c 2 x 2 (t)c 1 y 1 (t) + c 2 y 2 (t) h(t) (t) h(t) x(t) y(t) h(t)

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17 Lecture on Communication Theory

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18 Lecture on Communication Theory x(t) y(t) 1 1 0 남자 여자 0 * correlation T t T t * T * * T t 2T t -2T T 2T t t

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19 Lecture on Communication Theory ex12) Tapped-Delay-line Filter assumptions) h(t) = 0 for t < 0 h(t) = 0 for t T f sampling with , t=n , =k where N = T f let w k = h(k ) y(n ) = = w 0 x[n ] + w 1 x[n - ] +..... + w N-1 x[n -(N-1) ] 2 1

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20 Lecture on Communication Theory 2. Causality & stability 1) Causal if a system does not respond before the excitation is applied h(t) = 0, t < 0 causality Real time 으로 동작 하는 system must be causal Memory 기능이 있는 것 can be non-causal 2) Stable if the output signal is bounded for all bounded input signals (BIBO) cf, BIBO: bounded input-bounded output BIBO stability criterion BIBO stability 3. Frequency Response 1) y(t) = x(t) h(t) Y(f) = X(f)H(f) h(t) x(t) y(t)

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21 Lecture on Communication Theory 2) H(f) 의 (freq. domain 에서의 ) 표현 H(f) = |H(f)| exp(j (f)) |H(f)| : Amplitude response (f) : phase response h(t) 가 real H(f) : conjugate symmetry |H(f)| = |H(-f)| even (f) = - (-f) odd polar form ln H(f) = ln |H(f)| + j (f) = (f) + j (f) (f):gain[nepers] (f):[radians] ) ’(f) = 20 log 10 |H(f)| [dB] decibels ’(f) = 8.69 (f) 1 neper = 8.69 dB 3) BW - 3 dB BW 4. Paley-Wiener Criterion : freq. domain equivalent of causality requirement (f) : gain of causal filter 0 -B B -f c f c -B f c +B

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22 Lecture on Communication Theory 1. Ideal Low Pass Filter |H(f)| = 1, -B f B 0, |f| > B h(t) = 2B sinc[2B(t-t 0 )] (f) = -2 f t 0 : linear phase To make a causal filter, | sinc[2B(t-t 0 )] | << 1 for t < 0 If making digital filter, non-causal is O.K. 2. computer experiment I pulse response of ideal LPF 2.9 Filters -T/2T/2 X(t) H(f) f -B B 5/T 1/T 10/T BT=10 B=5/T, BT=5 1/B t0t0 t 0 =0 t 0 >0

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23 Lecture on Communication Theory (observations) overshoot = 9 % Gibb’s phenomenon overshoot is independent of B frequency of ripple is B 3. Fig 27. B = 1 Hz 일 때 f 0 = 0.1(T=5), 0.25(T=2), 0.5(t=1), 1(T=0.5) Hz 입력이 들어갔을 때의 output conclusion) BT 1 이 되어야 recognizable output 이 나온다 4. Design of Filters 1) Basic Design steps approximation of a prescribed frequency response by a realizable transfer function Realization of the approximating transfer function by a physical device Ringing BT oscillation freq. 5 5 Hz 1010 Hz 2020 Hz 100100 Hz overshoot (%) 9.11 8.98 8.99 9.63

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24 Lecture on Communication Theory 2) Stable system : BIBO complex frequency s = j2 f plane 상에서 표시 H’(s) is a rational function H’(s) = H(f) | j2 f=s z 1 z 2,... z m : zeros p 1 p 2,... p n : poles stability Re[p i ] < 0 for all i Minimum-phase systems Re[p i ] < 0, Re[z i ] < 0 Non-minimum phase systems Re[p i ] < 0, - Re[z i ] 3) Butterworth filters Chebyshev filters 4) Implementation of filters Analog filters : (a) L, C (b) C, R, OP-amp Discrete-time filters : switched-capacitor filters (SAW) surface accoustic wave filters Digital filters : FIR, IIR ( 장점 ) programmable, flexibility SRSR SISI =

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25 Lecture on Communication Theory 2.10 Hilbert Transform 1. Frequency selective filters Phase selective filters 180º : ideal transformer ± 90 º : Hilbert Transform 2. Def. of Hilbert Transform f sgn(f) And 1/ t -jsgn(f) 1 3. Applications 1) SSB Modulation 2) Mathematical basis for the representation of band-pass signals

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26 Lecture on Communication Theory ex13) 4. Properties of Hilbert transform Assumption) g(t) is real-valued property 1) & have the same amplitude spectrum 2) 3) pf) f G(f) f jG(f) x 0

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27 Lecture on Communication Theory 2.11 Pre - envelope 1. Pre-envelope for positive frequency. For a read-valued signal g( t) pre-envelope of g(t) 2. Pre-enveloped for negative frequency. -W W G(f) f G + (f) -W W f

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28 Lecture on Communication Theory 3. 용도 : Useful in handling band-pass signals and systems 2.12 Canonical Representations of Band-Pass signals 1. Base-band 신호와 Pass-band 신호의 관계 Base-band signal ( 복소수 ) : complex envelope Pass-band signal ( 실수 ) : band-pass signal, narrow-band signal -W W G(f) f G - (f) -W W f Pre-envelope for negative freq. Re( )

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29 Lecture on Communication Theory Pre-envelope of g(t) physical line 2 line( 복소수 ) 1 line( 실수 ) spectral efficiency is same

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30 Lecture on Communication Theory 2. Physical Implementation of pass-band signal

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31 Lecture on Communication Theory 3. Expression in the polar form Pass-Band 4. Envelope 용어 정리

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32 Lecture on Communication Theory 0 0 f -f c fcfc 0 fcfc ex14) t -T/2T/2 g + (t) real g + (t) image g(t)

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33 Lecture on Communication Theory 2.13 band-pass systems 1.Pass-band 신호를 Base-band 에서 표현

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34 Lecture on Communication Theory 2. Complex Filter 의 구현

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35 Lecture on Communication Theory 3. Band-pass system 의 response 를 구하는 과정 summary 4. Computer Experiment II. Response of Ideal Base-pass Filter to a pulsed RF Wave BT=5 일 경우의 RF 파형 Base-band 파형 + Modulation

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36 Lecture on Communication Theory 2.14 Phase and Group Delay A dispersive channel in a polar form

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37 Lecture on Communication Theory

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38 Lecture on Communication Theory 2.15 Numerical Computation of the Fourier Transform 1. DFT & IDFT 1) Nyquist Sampling Theorem Sampling Rate should be greater than twice the highest frequency component of the input signal to avoid aliasing. 2) DFT & IDFT Given finite data sequence {g 0,g 1,,g N-1 } ex) analog g n =g(nT s ) DFT of g n w here k=0,1,,N-1 IDFT of G k w here n=0,1,,N-1 {G 0,G 1,,G N-1 } : transform sequence k=0,, N-1 : frequency index t T T=NT s f s =N f fsfs fsfs

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39 Lecture on Communication Theory 3) Physical meaning of DFT 2. Interpretation of the DFT and the IDFT. DFT & IDFT: G k and g n must be periodic. DFT & IDFT: are linear. DC 2cycle/N 1cycle/N

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40 Lecture on Communication Theory 3.FFT Algorithms. DFT of g n i.e. W kn : periodic with period N. Assume N=2 L, where L is integer (1) , (1) k = 0, 1, 2,,,,,,,N-1 where

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41 Lecture on Communication Theory N point FFT = 2 [ -point FFT] 1 2 ; N/2-point FFT

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42 Lecture on Communication Theory ex) Fig 2.38 Butterfly. ex) Fig 2.38

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43 Lecture on Communication Theory # of computation of DFT N 2 complex(x)+N(N-1)complex(+) # of computation of FFT if decimation in freq. one complex(x) + 2complex(+) ex) N=1024=1k DFT 1M FFT 5K 단 FFT 는 2 N 개로 해야 함 Other algorithm : Decimation-in-time algorithm 4. Computation of IDFT 200 배 차이 N log 2 N computations complex conjugate FFT complex conjugate N where : complex conjugate

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44 Lecture on Communication Theory 2.30, 2.32 문제 ) N=16(g 0 ~g 15 ) 일 때의 Decimation-in-freq. Algorithm 을 그림으로 그려라. 2.1 (see book) Bit reversed index.

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S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communications Systems ECE.09.331 Spring 2007 Shreekanth Mandayam ECE Department Rowan University.

S. Mandayam/ ECOMMS/ECE Dept./Rowan University Electrical Communications Systems ECE.09.331 Spring 2007 Shreekanth Mandayam ECE Department Rowan University.

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