Chapter 2. Fourier Representation of Signals and Systems

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

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

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

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]

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

2.7 Ideal Low-Pass Filters 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

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

2.7 Ideal Low-Pass Filters Gibbs phenomenon

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

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.

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

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)

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

Summary Fourier Transform Inverse relationship Linear filtering 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