Chapter 2. Fourier Representation of Signals and Systems

1. Chapter 2. Fourier Representation of Signals and Systems

2. 2.6 Transmission of Signal Through Linear Systems : Convolution Revisited • In a linear system, • The response of a linear system to a number of excitations applied simultaneously is equal to the sum of the responses of the system when each excitation is applied individually. • Time Response • Impulse response • The response of the system to a unit impulse or delta function applied to the input of the system. • Summing the various infinitesimal responses due to the various input pulses, • Convolution integral • The present value of the response of a linear time-invariant system is a weighted integral over the past history of the input signal, weighted according to the impulse response of the system

3. 2.6 Transmission of Signal Through Linear Systems : Convolution Revisited • Causality and Stability • Causality : It does not respond before the excitation is applied • Stability • The output signal is bounded for all bounded input signals (BIBO) • An LTI system to be stable • The impulse response h(t) must be absolutely integrable • The necessary and sufficient condition for BIBO stability of a linear time-invariant system

4. 2.6 Transmission of Signal Through Linear Systems : Convolution Revisited • Frequency Response • Impulse response of linear time-invariant system h(t), • Input and output signal • By convolution theorem(property 12), • The Fourier transform of the output is equal to the product of the frequency response of the system and the Fourier transform of the input • The response y(t) of a linear time-invariant system of impulse response h(t) to an arbitrary input x(t) is obtained by convolving x(t) with h(t), in accordance with Eq. (2.93) • The convolution of time functions is transformed into the multiplication of their Fourier transforms

5. 2.6 Transmission of Signal Through Linear Systems : Convolution Revisited • In some applications it is preferable to work with the logarithm of H(f) Amplitude response or magnitude response Phase or phase response The gain in decible [dB]

6. 2.6 Transmission of Signal Through Linear Systems : Convolution Revisited • Paley-Wiener Criterion • The frequency-domain equivalent of the causality requirement

7. 2.7 Ideal Low-Pass Filters • Filter • A frequency-selective system that is used to limit the spectrum of a signal to some specified band of frequencies • The frequency response of an ideal low-pass filter condition • The amplitude response of the filter is a constant inside the passband -B≤f ≤B • The phase response varies linearly with frequency inside the pass band of the filter

8. 2.7 Ideal Low-Pass Filters • Evaluating the inverse Fourier transform of the transfer function of Eq. (2.116) • We are able to build a causal filter that approximates an ideal low-pass filter, • with the approximation improving with increasing delay t

9. 2.7 Ideal Low-Pass Filters • Gibbs phenomenon

10. 2.8 Correlation and Spectral Density : Energy Signals • The autocorrelation function of an energy signal x(t) is defined as

11. 2.8 Correlation and Spectral Density : Energy Signals • Energy spectral density • The energy spectral density is a nonnegative real-valued quantity for all f, even though the signal x(t) may itself be complex valued. • Wiener-Khitchine Relations for Energy Signals • The autocorrelation function and energy spectral density form a Fourier-transform pair.

12. 2.8 Correlation and Spectral Density : Energy Signals • Cross-Correlation of Energy Signals • The cross-correlation function of the pair • The energy signals x(t) and y(t) are said to be orthogonal over the entire time domain • If Rxy(0) is zero • The second cross-correlation function

13. 2.8 Correlation and Spectral Density : Energy Signals • The respective Fourier transforms of the cross-correlation functions Rxy(τ) and Ryx(τ) • With the correlation theorem • The properties of the cross-spectral density • Unlike the energy spectral density, cross-spectral density is complex valued in general. • Ψxy(f)= Ψ*yx(f) from which it follows that, in general, Ψxy(f)≠ Ψyx(f)

14. 2.9 Power Spectral Density • The average power of a signal is • Power signal : • Truncated version of the signal x(t) • By Rayreigh energy theorem Power spectral density

15. Summary • Fourier Transform • A fundamental tool for relating the time-domain and frequency-domain descriptions of a deterministic signal • Inverse relationship • Time-bandwidth product of a energy signal is a constant • Linear filtering • Convolution of the input signal with the impulse response of the filter • Multiplication of the Fourier transform of the input signal by the transfer function of the filter • Correlation • Autocorrelation : a measure of similarity between a signal and a delayed version of itself • Cross-correlation : when the measure of similarity involves a pair of different signals • Spectral Density • The Fourier transform of the autocorrelation function • Cross-Spectral Density • The Fourier transform of the cross-correlation function