Representation of Continuous and Discrete-Time Signals-2 Study notes

Operations in Continuous Time Signals:

Periodic & Non-Periodic Signals:

A signal is a periodic signal if it completes a pattern within a measurable time frame, called a period and repeats that pattern over identical subsequent periods.
The period is the smallest value of T satisfying g(t + T) = g(t) for all t. The period is defined so because if g(t + T) = g(t) for all t, it can be verified that g(t + T') = g(t) for all t where T' = 2T, 3T, 4T, ... In essence, it's the smallest amount of time it takes for the function to repeat itself. If the period of a function is finite, the function is called "periodic".
Functions that never repeat themselves have an infinite period, and are known as "aperiodic functions".

Even & Odd Signals:

A function even function if it is symmetric about the y-axis. While, A signal is odd if it is inversely symmetrical about the y-axis.

Even Signal, f(x) = f(-x)

Odd Signal, f(x) = - f(-x)

Note: Some functions are neither even nor odd. These functions can be written as a sum of even and odd functions. A function f(x) can be expressed in terms of sum of an odd function and an even function.

Invertibility and Inverse Systems:

A system is invertible if distinct inputs results distinct outputs. As shown in the figure for the continuous-time case, if a system is invertible, then an inverse system exists that, when cascaded with the original system, results an output w(t) equal to the input x(t) to the first system.

An example of an invertible continuous-time system is y(t) = 2x(t),

for which the inverse system is w(t) = 1/2 y(t)

Causal System:

A system is causal if the output depends only on the input at the present time and in the past. Such systems are often referred as non anticipative, as the system output does not anticipate future values of the input. Similarly, if two inputs to a causal system are identical up to some point in time t_o or n_o the corresponding outputs must also be equal up to this same time.

y₁(t) = 2x(t) + x(t-1) + [x(t)]²⇒ Causal Signal

y₁(t) = 2x(t) + x(t-1) + [x(t+2)] ⇒ Non-Causal Signal

Homogeneity (Scaling):

A system is said to be homogeneous if, for any input signal X(t), i.e. When the input signal is scaled, the output signal is scaled by the same factor.

Time-Shifting / Time Reversal / Time Scaling

Time Shifting can be understood as shifting the signal in time. When a constant is added to the time, we obtain the advanced signal, & when we decrease the time, we get the delayed signal.

Time Scaling:

Due to the scaling in time the output Signal may shrink or stretch it depends on the numerical value of scaling factor.

Time Inversion:

Time Inversion referred as flipping the signal about the y-axis.

The Correlation Functions

Correlation is a mathematical operation that is similar to convolution. As convolution, in correlation two signals are used to produce a third signal. The third signal is referred to as the cross-correlation of the two input signals. When a signal is correlated with itself, the resultant signal is called autocorrelation.

There are three basic definitions to define correlation function

(a) For an infinite duration waveform:-

It can be considered as a “power” based definition.

(b) For the finite duration waveform:- If the waveform exists only in the interval t₁ ≤t ≤ t_2.

It cab be considered as an “energy” based definition.

(c) For the periodic waveform:- f(t) is periodic with period T then.

for an arbitrary t₀, it again can be considered as a “power” based definition.

Example:- Obtain the auto-correlation function of the square pulse which have amplitude a and duration T as shown in figure below.

The wave form has a finite duration, and the auto-correlation function is

The auto-correlation function is developed graphically below

Properties of the Auto-correlation Function

The auto-correlation functions φff (τ ) and ρff (τ ) are even functions, that is

φ_ff (−τ ) = φ_ff (τ ), and ρ_ff (−τ ) = ρ_ff (τ )

A maximum value of ρ_ff(τ ) (or φ_ff(τ ) occurs at delay τ = 0,

|ρ_ff(τ )| ≤ ρ_ff (0), and |φ_ff (τ )| ≤ φ_ff(0)

and we note that is the “energy” of the waveform.

Similarly

ρ_ff (τ) contains no phase information, and is independent of the time origin.
If f(t) is periodic with period T, φ_ff (τ) is also periodic with period T.
If f(t) has zero mean (µ = 0), and f(t) is non-periodic, lim ρ_ff (τ ) = 0.

Note on the relative “widths” of the Auto-correlation and Power/Energy Spectra

As in the case of Fourier analysis of waveforms, there is a general reciprocal relationship between the width of a signal's spectrum and the width of its auto-correlation function. A narrow autocorrelation function generally implies a “broad” spectrum

and a “broad” autocorrelation function generally implies a narrow-band waveform.

From the limit, if φ_ff (τ)= δ(τ), then Φ_ff (j Ω)=1, and the spectrum is said to be “white”.

The Cross-Correlation Function

The cross-correlation is referred as a measure of self-similarity between two waveforms f(t) and g(t). As in the case of the auto-correlation functions we need two definitions:

in the case of infinite duration waveforms, and

for finite duration waveforms.

Example:- Find the cross-correlation function between the two functions as shown in figure.

It is clear in figure that g(t) is a delayed version of f(t). The cross-correlation is

where the peak occurs at τ = T₂−T₁ (the delay between the two signals).

Properties of the Cross-Correlation Function

φ_fg(τ ) = φ_gf (−τ), and the cross-correlation function is not necessarily an even function.
If φ_fg(τ ) = 0 for all τ , then f(t) and g(t) are said to be uncorrelated.
If g(t) = a.f(t−T), where a is a constant, that is g(t) is a scaled and delayed version of f(t), then φ_ff(τ) will have its maximum value at τ = T.
Cross-correlation is often used in optimal estimation of delay, such as in echolocation (radar, sonar), and in GPS receivers.

The Cross-Power/Energy Spectrum

We define the cross-power/energy density spectra as the Fourier transforms of the cross-correlation functions:

R_fg(jΩ) = (F−jΩ).G(jΩ)

Note that although R_ff (jΩ) is real and even because ρ_ff(τ) is real and even, this is not the case with the cross-power/energy spectra,Φ_fg(jΩ) and R_fg(jΩ),and they are in general complex.

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