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USEFUL FORMULAS Properties of valid Autocorrelation function (ACF) The ACF of a signal is defined as the measure of similarity or coherence between a signal and ts time delayed version. • ACF of a real function x(t): Rxx(T) = Ext) xx(t+T)), where T is a variable called time-shift (delay) parameter • ACF is an even function • • ACF reaches its peak at the origin: R(T) R(0) ACF of a periodic function is, itself, periodic with the same period Autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all) is the sum of the autocorrelations of each function separately • ACF is related to the power spectral density via the Fourier transform Rxx (T→ 0) = m₂(x); Rxx (T→ Infinity) = [m₂(x)]2 O Choice 1 O Choice 2 O Choice 3 O Choice 4 O Choice 5

USEFUL FORMULAS Properties of valid Autocorrelation function (ACF) The ACF of a signal is defined as the measure of similarity or coherence between a signal and ts time delayed version. • ACF of a real function x(t): Rxx(T) = Ext) xx(t+T)), where T is a variable called time-shift (delay) parameter • ACF is an even function • • ACF reaches its peak at the origin: R(T) R(0) ACF of a periodic function is, itself, periodic with the same period Autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all) is the sum of the autocorrelations of each function separately • ACF is related to the power spectral density via the Fourier transform Rxx (T→ 0) = m₂(x); Rxx (T→ Infinity) = [m₂(x)]2 O Choice 1 O Choice 2 O Choice 3 O Choice 4 O Choice 5

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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USEFUL FORMULAS Properties of valid Autocorrelation function (ACF) The ACF of a signal is defined as the measure of similarity or coherence between a signal and its time delayed version. • ACF of a real function x(t): Rxx(T) = Ex(t) x x(t+T)], where T is a variable called time-shift (delay) parameter ACF is an even function ACF reaches its peak at the origin: R(T) = R(0) ACF of a periodic function is, itself, periodic with the same period Autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all) is the sum of the autocorrelations of each function separately ACF is related to the power spectral density via the Fourier transform Rxx (T-0) = m₂(x); Rxx (T→ Infinity) = [m₁(x)]² O Choice 1 O Choice 2 Choice 3 Choice 4 O Choice 5
Transcribed Image Text:USEFUL FORMULAS Properties of valid Autocorrelation function (ACF) The ACF of a signal is defined as the measure of similarity or coherence between a signal and its time delayed version. • ACF of a real function x(t): Rxx(T) = Ex(t) x x(t+T)], where T is a variable called time-shift (delay) parameter ACF is an even function ACF reaches its peak at the origin: R(T) = R(0) ACF of a periodic function is, itself, periodic with the same period Autocorrelation of the sum of two completely uncorrelated functions (the cross-correlation is zero for all) is the sum of the autocorrelations of each function separately ACF is related to the power spectral density via the Fourier transform Rxx (T-0) = m₂(x); Rxx (T→ Infinity) = [m₁(x)]² O Choice 1 O Choice 2 Choice 3 Choice 4 O Choice 5
Check with reasons whether the functions f(t, t+T) listed below represent in each case a valid autocorrelation function (ACF); if so, why? Case (2-ITI) f(tt+T) (-2sTs+2) O otherwise pi= 3.1416 Choices 1 2 3 4 5 (1) Valid ACF? YES/NO No: Not maximum at T=0 Odd function No: (ii) Reason(s)? Minimum at T=0 Yes; (axt + (1 - (TI) (-1 sTs+1) and for all t O otherwise a: constant Yes; Valid Yes: Maximum at T = Ely) is 0 Yes; It is an odd Even symmetry of time (ii) function 0 No: It is a function of time independent No: Maximum at T = It is a function O with even of time symmetry No: No: Maximum at T = It is a function of time. (iii) (AT+B)/(C|T|+D) (A, B, C, D): Constants and B/D > A/C (iii) No: It is not an even function Yes: Valid Max at T = 0 Even symmetry It is not function Maximum of time No: at T=0 Yes; Even function Maximum at T=0 Yes: It is not al function of time sin(|T) + 1 (-pi/2 s T s +pi/2) 0 otherwise (iv) Yes; Yes; Maximum at T Maximum at T = 0 =0 with No: It is a function (v) Even symmetry function of time (v) Yes; Valid Yes; Valid ACF Even symmetry Not a function of at the ordinate time. Even function No: Minimum at T = 0 It is an even function axsin(t) + bxT (- pi/2 sTs+ pi/2) 0 otherwise No: It is a function of time It is an even Yes: Maximum at T = 0 It has odd symmetry No: It is a function of time Yes; It is not a function of time
Transcribed Image Text:Check with reasons whether the functions f(t, t+T) listed below represent in each case a valid autocorrelation function (ACF); if so, why? Case (2-ITI) f(tt+T) (-2sTs+2) O otherwise pi= 3.1416 Choices 1 2 3 4 5 (1) Valid ACF? YES/NO No: Not maximum at T=0 Odd function No: (ii) Reason(s)? Minimum at T=0 Yes; (axt + (1 - (TI) (-1 sTs+1) and for all t O otherwise a: constant Yes; Valid Yes: Maximum at T = Ely) is 0 Yes; It is an odd Even symmetry of time (ii) function 0 No: It is a function of time independent No: Maximum at T = It is a function O with even of time symmetry No: No: Maximum at T = It is a function of time. (iii) (AT+B)/(C|T|+D) (A, B, C, D): Constants and B/D > A/C (iii) No: It is not an even function Yes: Valid Max at T = 0 Even symmetry It is not function Maximum of time No: at T=0 Yes; Even function Maximum at T=0 Yes: It is not al function of time sin(|T) + 1 (-pi/2 s T s +pi/2) 0 otherwise (iv) Yes; Yes; Maximum at T Maximum at T = 0 =0 with No: It is a function (v) Even symmetry function of time (v) Yes; Valid Yes; Valid ACF Even symmetry Not a function of at the ordinate time. Even function No: Minimum at T = 0 It is an even function axsin(t) + bxT (- pi/2 sTs+ pi/2) 0 otherwise No: It is a function of time It is an even Yes: Maximum at T = 0 It has odd symmetry No: It is a function of time Yes; It is not a function of time
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