Presented By,
Shahdadpuri Sunny
Bhavsar Ripal
Prajapati Sahil
Soni Akshit
Mangukia Saurabh

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What is Spectral density??
Spectral density is a positive real function of a frequency variable associated with a stationary stochastic process, or a deterministic function of time.
Spectral density has dimension of power per HZ or energy per HZ.
The spectral densities are flat, i.e. independent of frequency.
Flat spectral density:-
• Spectral density is often called the spectrum of the signal .
The spectral density captures the frequency content of a stochastic process and help identify periodicities.
Thermal noise falls into the category of power signals, and hence it has a spectral density.
The spectral density of a signal, defines the distributions of energy or power per unit bandwidth as a function of frequency.
Spectral density: Period gram
• spectral density: period gram
Types of spectral density
Mainly three types of spectral density.
1. Energy spectral density (PDS)
2. Power spectral density (EDS)
3. Cross spectral density (CDS)
Energy spectral density (EDS)
The energy spectral density describes how the energy of a signal or a time series is distributed with frequency.
If f(t) is a finite-energy signal, the spectral density Φ(ω) of the signal is the square or the magnitude of the continuous Fourier transformation of the signal.
Here energy is taken as the integral of the square of the signal.
• The equation of EDS is,
Where, ω = angular frequency
F(ω) = continuous Fourier transformation
of f(t)
F*(ω) = complex conjugate
If the signal is discrete with values fn, over an infinite number of elements, we still have an energy spectral density
If the number of defined values is finite, the sequence does not have an energy spectral density as per sec, but the sequence can be treted as periodic using a discrete Fourier transform to make discrete spectrum.
It can be extended with zeros and a spectral density can be compute as in the infinite-sequence case.
Its unit is (Voltage)2 per HZ.
• Energy spectral wave:
• Signal energy:-
The energy of signal is the shaded region.
Signal as a function of varying amplitude with time.
So, the strength of the signal would be area under the tuned, however this area may be negative.
the equation is,
Power spectral density (PDS)
The power spectral density is describes how the power of signal or time series is distributed with frequency.
Here power spectral density response shown,
Here f1 is δPn1 = Sp(f1)δf.
At frequency f1. the available noise power for an infinitesimally small bandwidth δf.
This is so because the bandwidth del f may be assumed flat about f1.
The available power is given as the product of spectral density (watts/hertz) x bandwidth (HZ).
The unit of PDS is watts/HZ.
Power spectrum waves:
• Power signal:-
If the signal does not decayed, in this case we have infinite energy for any such signal.
This lead to the idea of signal power.
so, its given by,
All the energy signals are not the power signals.
Cross spectral density (CDS)
The power spectral density is the Fourier transform of the auto-covariance function we may define the cross spectral density (CDS) as the Fourier transform of the cross-covariance function.
The Coherence (signal processing) for use of the cross-spectral density.
Cross spectral density:
• Properties :-
The spectral density of f(t) and the autocorrelation of f(t) from a Fourier transform pair.
One of the result of Fourier analysis is parseval’s theorem which states that the area under the energy spectral density curve is equal t the area under the square of the magnitude of the signal, the total energy:
Properties :-
The spectral density of f(t) and the autocorrelat -ion of f(t) from a Fourier transform pair.
One of the result of Fourier analysis the area under the energy spectral density curve is equal t the area under the square of the magnitude of the signal, the total energy:
Equation,
Applications:-
The concept and use of the power spectrum of a signal is fundamental in electronic communication system engineering.
E.g.:- Radio & microwave communication, radar, and related system.
spectrum analyzer is use for aiding electronics engineers, technologist, & technicians in observing & measuring the power spectrum of electronic signals.
STFT is good smooth estimate of its PDS.
Radar spectrum:
• The spectrum analyzer measures essentially the magnitude of the short-time Fourier transformation (STFT) of an input signal.
Spectrum analyzer
Conclusion:-
The spectral density is a function of a frequency.
The spectral density is usually estimated using Fourier transform method.
Its generated in Thermal noise & its further initialize in terms of Energy & Power.