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
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 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