Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev

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Electrical Engineering (EE) : Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev

The document Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev is a part of the Electrical Engineering (EE) Course Signal and System.
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Operations in Continous 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".
    Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev

Even & Odd Signals:
A function is said to be even function if it is symmetric about the y-axis. While on the other hand, 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)
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev
Note: Some functions are neither even nor odd. However, such functions can be written as a sum of even and odd functions. Any function f(x) can be expressed as a sum of an odd function and an even function.
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev

Invertibility and Inverse Systems:
A system is said to be invertible if distinct inputs lead to distinct outputs. As illustrated 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, yields an output w(t) equal to the input x(n) 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)
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev

Causal System:
A system is causal if the output at any time depends only on values of the input at the present time and in the past. Such a system is often referred to as being nonanticipative, as the system output does not anticipate future values of the input. Consequently, if two inputs to a causal system are identical up to some point in time to or no 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. scaling any input signal scales the output signal by the same factor.
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev

Time-Shifting / Time Reversal / Time Scaling 
Time Shifting is simply shifting the signal in time. When we add a constant to the time, we obtain the advanced signal, & when we decrease the time, we get the delayed signal.
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev

Time Scaling: 
Due to the scaling in time the output Signal may shrink or strech it depends on the numerical value of scaling factor.
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev

Time Inversion:
Time Inversion is simply flipping the signal about the y-axis.
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev

The Correlation Functions
Correlation is a mathematical operation that is very similar to convolution. Just as with convolution, correlation uses two signals to produce a third signal. This third signal is called the cross-correlation of the two input signals. If a signal is correlated with itself, the resulting signal is instead called the autocorrelation.
There are three basic definitions to define correlation function
(a) For an infinite duration waveform:
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev
which may be considered as a “power” based definition.
(b) For a finite duration waveform: If the waveform exists only in the interval t1 ≤t ≤ t2.
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev
which may be considered as an “energy” based definition.
(c) For a periodic waveform: If f(t) is periodic with period T.
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev
for an arbitrary t0, which again may be considered as a “power” based definition.
Example: Find the auto-correlation function of the square pulse of amplitude a and duration T as shown below.
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev
The wave form has a finite duration, and the auto-correlation function is
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev
The auto-correlation function is developed graphically below
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev

Properties of the Auto-correlation Function

  • The auto-correlation functions φff (τ) and ρff (τ) are even functions, that is
    φff (−τ) = φff (τ), and ρff (−τ) = ρff (τ)
  • A maximum value of ρff(τ) (or φff(τ) occurs at delay τ = 0,
    ff(τ)| ≤ ρff (0), and |φff (τ)| ≤ φff (0)
    and we note that Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev is the “energy” of the waveform.

Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev

  • ρff (τ) contains no phase information, and is independent of the time origin.
  • If f(t) is periodic with period T, φff (τ) is also periodic with period T.
  • If f(t) has zero mean (µ = 0), and f(t) is non-periodic, lim ρff (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
    Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRevand a “broad” autocorrelation function generally implies a narrow-band waveform.

Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev
In the limit, if φff (τ)= δ(τ), then Φff (j Ω)=1, and the spectrum is defined to be “white”.
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev

The Cross-Correlation Function
The cross-correlation function is 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:
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev
in the case of infinite duration waveforms, and
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev for finite duration waveforms.
Example: Find the cross-correlation function between the following two functions.
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev
In this case g(t) is a delayed version of f(t). The cross-correlation is
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev
where the peak occurs at  τ = T2 − T1 (the delay between the two signals).

Properties of the Cross-Correlation Function

  • φfg(τ) = φgf (−τ), and the cross-correlation function is not necessarily an even function.
  • If φfg(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 φff(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:
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev
Representation of Continuous and Discrete, Time Signals (Part -2) Electrical Engineering (EE) Notes | EduRev
Rfg (jΩ) = (F−jΩ).G(jΩ)
Note that although Rff (jΩ) is real and even (because ρff(τ) is real and even,this is not the case with the cross-power/energy spectra,Φfg (jΩ) and Rfg(jΩ),and they are in general complex.

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