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Theory of Signals and Systems Assignment

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Added on  2020-04-01

Theory of Signals and Systems Assignment

   Added on 2020-04-01

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Running head: Signals and systems1Theory of Signals and SystemsAuthorCourseDate
Theory of Signals and Systems Assignment_1
Signals and systems2Correlation functionsCorrelation functions find application in time domain specification of signals. The autocorrelation function of a signal h(t) is expressed by the following equation:Rh (τ) = limT→∞1T∫−T2T2h(t)h(t+τ)dtThe average of the product h(t)h(t+τ) is dependent on the value of τ. As the correlation function aims at finding similarity of the components in the product,Rh (τ) = 0 whenh(t)∧h(t+τ) have no similarity and each has a mean value of zero. Correlation can therefore be used to capture similar data patterns or in matching signals whereas autocorrelation is can be applied to decipher repetitive patterns in a signal and recognizing noise.The power spectral density of a signal is computed for all time over which the signal exists whilethe energy spectral density is computed around the finite time for a signal concentrated around a finite time interval.The energy of a signal is expressed as:E = ∫−∞∞¿x(t)∨¿¿ ^2 dtSince energy spectral density is for signals with finite total energy, the Parseval’s theorem as depicted in [3] gives the expression of the energy as:
Theory of Signals and Systems Assignment_2
Signals and systems3∫−∞∞¿x(t)∨¿¿ ^2 dt = ∫−∞∞¿ẋ(t)∨¿¿ ^2 dfwhere the latter part of the equation is the Fourier transform of the signal.The power spectral density on the other hand is the Fourier transform of the signals autocorrelation function. The average power is expressed as:P = limT→∞12T∫−TT∨x(t)∨¿^2 dt The Fourier transform for most signals do not exist in frequency analysis. Hence a truncated Fourier transform is used whereby a signal is integrated over an interval which is finite [0, T]. The PSD is therefore:S(ω) = limT→∞E [¿ẋ(ω)∨¿^2]where the expected value E as shown in [3] is:E[¿ẋ(ω)∨¿^2] = 1T∫0T∫0TE [x*(t)x(t’)] eiω(t−t')dt dt’Wiener-Khinchine theoremThe above expression relates to the autocorrelation function. The Wiener-Khichin theorem statesthat a random process included in finite power, the spectral analysis extends to determine the spectral distribution of power. Hence a Fourier transform pair is formed by the autocorrelation function and spectral power density as shown in [4]. This theory is given as:
Theory of Signals and Systems Assignment_3

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