Spectral concentration of power

Definition

One defines the spectral concentration of power ( DSP in summary, Power Spectral Density or PSD in English) as being the square module of the Transformée of Fourier . Thus, if X is a signal and X its transform of Fourier, the spectral concentration of power is worth $\ Gamma_x=|X|^2$ .

Spectral concentration of power and Autocorrelation

The definition of the function of temporal Autocorrélation of a signal X at continuous time is :

$\ gamma (\ tau) = \ int_ {- \ infty} ^ {+ \ infty} x^ {*} (T) X (t+ \ tau) \, dt$

where ^* is the conjugation complexes.

Taken at the point τ, this function measures to some extent the way in which the structures which one can see in a signal repeat on scales of time about τ.

Its definition using the operator of convolution is:

$\ gamma (\ tau) =x \ ast x^* (-)$

The properties of the Transformée of Fourier imply that the spectral concentration is the transform of Fourier of the autocorrelation:

$\ mathcal {F} = \ mathcal {F} \ times \ mathcal {F} =X \ cdot X^*= \ Gamma$

Detailed calculation

Let us calculate its transform of Fourier Γ (ν) :

$\ Gamma (\ jmath \ Omega) = \ int_ {- \ infty} ^ {+ \ infty} \ int_ {- \ infty} ^ {+ \ infty} x^* (T) X (t+ \ tau) e^ {- \ jmath \ Omega \ tau} \, dt \, D \ tau$ , “ $\ jmath$ ” indicating the complex number of square equal to -1.

This expression can be put under the forme :

$\ Gamma (\ jmath \ Omega) = \ int_ {- \ infty} ^ {+ \ infty} \ left (\ int_ {- \ infty} ^ {+ \ infty} X (t+ \ tau) e^ {- \ jmath \ Omega (t+ \ tau)}D \ tau \ right) x^* (T) e^ {+ \ jmath \ Omega T} \, dt$

One carries out in the central integral the change of variable u=t+τ and it vient :

$\ Gamma (\ jmath \ Omega) = \ int_ {- \ infty} ^ {+ \ infty} \ left (\ int_ {- \ infty} ^ {+ \ infty} X (U) e^ {- \ jmath \ Omega U} of the \ right) x^* (T) e^ {+ \ jmath \ Omega T} \, dt$

That is to say encore :

$\ Gamma (\ jmath \ Omega) =X (\ jmath \ Omega) \ int_ {- \ infty} ^ {+ \ infty} x^* (T) e^ {+ \ jmath \ Omega T} \, dt$

One carries out the change of variable u=-t and one obtient :

$\ Gamma (\ jmath \ Omega) =X (\ jmath \ Omega) \ int_ {- \ infty} ^ {+ \ infty} x^ {*} (- U) e^ {- \ jmath \ Omega U} \, du$

One recognizes, in the second term, the Transformée of Fourier of x^* (- T) . However the transform of Fourier of x^* is worth X^* (- ν) , and the transform of Fourier of X (- T) is worth X (ν) . The second term is worth thus X^* (jω) , therefore Γ (jω) =X (jω) X^* (jω) =|X (jω)|² : the spectral concentration of power is also the transform of Fourier of the Autocorrélation.

See spectral Analysis for elementary considerations.

Use of the spectral concentration of power in the Telecommunications

In telecommunications, one must often treat random signals. However, one cannot calculate the transform of Fourier of a signal not entirely known. On the other hand, one can calculate the autocorrelation of a random signal known by his statistical properties. The spectral concentration of power , often, is thus used in telecommunications.

Let us consider, for example, the “white Bruit”. The noise is a random typical example of signal. Often, the value of the noise, at a given moment, is absolutely not correlated with the value of the noise at the other moments. That results in a function of autocorrelation of the noise equal to a impulse of Dirac (i.e. equalizes ad infinitum into 0, and 0 elsewhere). Tranformée of Fourier of an impulse of Dirac is the constant unit (the module is worth 1 whatever the frequency). One defines then, by “white vibration”, a noise of which the spectral concentration is constant according to the frequency. In telecommunications, one often regards the noises as being white, all at least in the band-widths of the studied systems.

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