Spectral density

The spectral power density is related to the frequency of the power of a signal in a frequency band of infinitesimal. This density has the dimension of power · time, the specification mainly in units of watts / Hz or dBm / Hz. If the power spectral density specified over the frequency spectrum, a power density spectrum ( LDS) or auto power spectrum results (English Power Spectral Density (PSD ) ). The integral of all the frequencies gives the total power of a signal. While the Fourier transform of stationary processes ( such as noise or single-frequency signals) is unlimited, to such signals with the help of the LDS can be analyzed quantitatively. The LDS is the display form of spectrum analyzers, with finitesimal here the power on, ie finally, small frequency intervals is given.

General and definitions

Since f for stationary processes (t ) in general neither the energy nor the Fourier transform in the classical sense exist, it stands to temporary shares for and to consider otherwise. Applies According to the formula of Plancherel

If the average signal power

Exists, there is also the right side of the above formula and as a spectral description of the performance you can see the Power Spectral Density define (if the limit exists ) as

For every finite is called the size of the periodogram of f It represents an estimate of the spectral power density, but its expectation value does not match (not expected true ) and its variance does not vanish even for arbitrarily large (not consistent).

Properties and calculation

To determine the power spectral density of the Wiener Khinchin theorem is widely used where it is passed through the Fourier transform of the temporal autocorrelation function of the signal:

It is

Is the autocorrelation function of the time signal f (t). For noise signals, generally for processes that ergodicity must be provided that allows properties of the random variables, such as the expected value to determine from a sample function. In practice, only a finite time window are considered, so you have to restrict the limits of integration. For a stationary distribution, the correlation function is no longer dependent on the time t.

The car power density spectrum is straight, real and positive. This means a loss of information which prevents a reversal of this procedure.

If a ( noise ) process with power spectral density through a linear, time-invariant system with transfer transfer function, we obtain at the output a power density spectrum of

The transfer function is quadratic in the formula, since the spectrum is a power reading.

The auto power spectrum can be represented as one-sided spectrum GXX ( f) (f 0). We then have:

And

Calculation methods is usually restricted to bandlimited signals (signals whose frequencies vanish for large LDS ), which allow a discrete representation ( Nyquist -Shannon sampling theorem ). Unbiasedness, consistent estimates of band-limited signals based on a modification of the periodogram, for example, the Welch method or Bartlett method. Estimates based on the autocorrelation function called correlogram methods such as the Blackman -Tukey estimate.

Application and units

Knowledge and analysis of the power spectral density of the useful signal and noise is substantially to determine the signal to noise ratio and to optimize the corresponding filter for noise suppression, for example in the noise.

The auto power spectrum can be used for statements about the frequency content of the analyzed signals. Spectrum analyzers examine the voltage signals. For the display in the performance specification of the termination resistor is required. Using spectrum analyzers can be but not in an infinitesimal spectral frequency band, only in a frequency interval of finite length, to determine. The spectral representation is obtained mean square spectrum ( MSS ), and their root RMS spectrum (English Root-Mean - Square). The length of the frequency interval is always indicated with and is called the resolution bandwidth (English Resolution Bandwidth, RBW short or BW) in [ Hz]. The conversion in decibels, as on measurement standardized, according as the conversion of RMS in accordance with which the two displays are numerically identical in decibels. As units are, inter alia, [DBm ], [ dBV ] RMS [V ], PK [V ] (of English. Peak) used. The information always refer to the used resolution bandwidth [Hz]. For example, a sinusoidal signal with a voltage gradient of V at a terminating resistor of 50 ohms an effective voltage of 30 dBm or dBV or 16.9897 7.0711 V ( RMS) or 10 V ( PK ) for each resolution bandwidth.

Examples

• When the correlation function is a delta function, it is called white noise, in this case, is constant.
• For the thermal noise, or more precisely the spectral noise power density, the following applies: N0 = kB · T. At 27 ° C it is 4:10 -21 J = 4:10 -21 W / Hz = -204 dBW / Hz = -174 dBm / Hz
• In the picture a MSS can be seen from the function with a uniformly distributed noise process ( quantization noise ) at a sampling rate of 44100 Hz and a resolution bandwidth of BW = 43.1 Hz, as it could for example come from a CD. The peak at about -3 dB represents the sine signal to the noise floor at about -128 dB. Since the performance data relating to the resolution bandwidth, one can read the SNR (note the logarithmic law, the multiplications in addition transformed). The read off from the image SNR is so close to the theoretically expected by law.