Spectral power density

The spectral power density indicates the frequency -related power of a signal in an infinitesimally small frequency band or wavelength range . In the frequency representation, this density has the dimension power · time ; it is usually specified in the units watt / hertz or dBm / Hz. In the wavelength representation, this density has the dimension power / length . ${\displaystyle S_{XX}(\omega )}$

Power density spectrum of a signal

If the spectral power density is specified over the frequency spectrum , a power density spectrum  ( LDS ) or car power spectrum ( English : Power Spectral Density  ( PSD ), also active power spectrum ) is created. The integral over all frequencies or wavelengths gives the total power of a signal. While the Fourier transformation of stationary processes (e.g. noise or monofrequency signals) is unlimited, they can be analyzed quantitatively with the help of the LDS.

The LDS is the display form of spectrum analyzers , whereby the power is specified over specified frequency intervals ( English : resolution bandwidth  ( RBW )).

General and definition

Since in general neither the energy nor the Fourier transform exist for stationary processes in the classical sense, it makes sense to consider time-limited components for and otherwise. ${\displaystyle f(t)}$ ${\displaystyle \|f\|_{2}^{2}}$ ${\displaystyle F{\big (}f{\big )}(\omega )}$ ${\displaystyle f_{T}(t)=f(t)}$ ${\displaystyle |t|\leq T}$ ${\displaystyle 0}$

According to Plancherel's formula, the following applies

${\displaystyle {\frac {1}{2T}}\int \limits _{\mathbb {R} }|f_{T}(t)|^{2}\mathrm {d} t={\frac {1}{2T}}\int \limits _{\mathbb {R} }|F(f_{T})(\omega )|^{2}\mathrm {d} \omega }$

If the mean signal power

${\displaystyle r_{XX}(0):=\lim \limits _{T\to \infty }{\frac {1}{2T}}\int \limits _{-T}^{T}|f(t)|^{2}\mathrm {d} t}$

exists, the right-hand side of the above formula also exists and as a spectral description of the power one can define the spectral power density (if the limit value exists) as

${\displaystyle S_{XX}(\omega ):=\lim \limits _{T\to \infty }{\frac {1}{2T}}|F(f_{T})(\omega )|^{2}}$

For each finite size the quantity is called the periodogram of . It represents an estimated value of the spectral power density, the expected value of which does not correspond (not true to expectations ) and the variance of which does not vanish even for any size ( inconsistent ). ${\displaystyle T}$ ${\displaystyle {\text{Per}}_{T}(\omega ):={\frac {1}{2T}}|F(f_{T})(\omega )|^{2}}$ ${\displaystyle f}$ ${\displaystyle S_{XX}(\omega )}$ ${\displaystyle T}$

Properties and calculation

According to the Wiener-Chintschin theorem , the power spectral density is often given as a Fourier transform of the temporal autocorrelation function of the signal: ${\displaystyle r_{xx}(t)}$

${\displaystyle S_{XX}(\omega )=F(r_{xx})(\omega )={\frac {1}{2\pi }}\int _{-\infty }^{\infty }r_{xx}(t)e^{-i\omega t}\mathrm {d} t}$

It is

${\displaystyle r_{xx}(t)=\lim _{T\to \infty }{\frac {1}{2T}}\int _{-T}^{T}f(\tau )\;{\overline {f(t+\tau )}}\mathrm {d} \tau }$

the autocorrelation function of the temporal signal . ${\displaystyle f(t)}$

For noise signals, generally for processes, the ergodicity must be assumed, which allows properties of the random variables such as the expected value to be determined from a model function. In practice, only a finite time window can be considered, which is why the integration limits have to be restricted. Only for a stationary distribution does the correlation function no longer depend on time . ${\displaystyle t}$

The car power density spectrum is even, real and positive. This means a loss of information which prevents this procedure from being reversed ( irreversibility ).

If a (noise) process with a power density spectrum is transmitted via a linear, time-invariant system with a transfer function , a power density spectrum of ${\displaystyle S_{XX}(\omega )}$ ${\displaystyle H(\omega )}$

${\displaystyle S_{YY}(\omega )=|H(\omega )|^{2}\cdot S_{XX}(\omega ).}$

The transfer function is included in the formula with the square, since the spectrum is a power quantity. This means that z. B. P = current times voltage and current and voltage are both multiplied by: Thus * current * * voltage = * current * voltage. ${\displaystyle |H(\omega )|}$ ${\displaystyle |H(\omega )|}$ ${\displaystyle |H(\omega )|}$ ${\displaystyle |H(\omega )|^{2}}$

The car performance spectrum can be represented as a one-sided spectrum with . Then: ${\displaystyle G_{XX}(f)}$ ${\displaystyle f\geq 0}$

${\displaystyle G_{XX}=S_{XX}(f)\quad \mathrm {f{\ddot {u}}r} \quad f=0}$

and

${\displaystyle G_{XX}=2S_{XX}(f)\quad \mathrm {f{\ddot {u}}r} \quad f>0\,.}$

Calculation methods are usually limited to band-limited signals (signals whose LDS disappears for high frequencies) that allow a discrete representation ( Nyquist-Shannon sampling theorem ). Faithful to expectation, consistent estimates of band-limited signals based on a modification of the periodogram are e.g. B. the Welch method or Bartlett method. Estimates based on the autocorrelation function are called correlogram methods, for example the Blackmann-Tukey estimation.

Application and units

The knowledge and analysis of the spectral power density of the useful signal and noise is essential for determining the signal-to-noise ratio and for optimizing appropriate filters for noise suppression , for example in image noise . The car power spectrum can be used for statements about the frequency content of the analyzed signals.

Spectrum analyzers examine the voltage of signals. For the display in power, the specification of the terminating resistor is required. Using spectrum analyzers, however, the spectral power cannot be determined in an infinitesimal frequency band, but only in a frequency interval of finite length. The spectral representation obtained in this way is called the mean square spectrum (MSS) and its root RMS spectrum (English. Root-Mean-Square).

The length of the frequency interval is always given and is called resolution bandwidth (Engl. Resolution Bandwidth , RBW short or BW) in the unit  Hz . The conversion to decibels is as standardized for performance data:

${\displaystyle {\text{MSS}}_{dB}=10\log _{10}({\text{MSS}})}$,

while the conversion for RMS is:

${\displaystyle {\text{RMS}}_{dB}=20\log _{10}({\text{RMS}})}$

This means that the two displays in decibels are numerically identical.

The units are u. a. used:

• dBm
• dBV
• RMS- [V]
• PK- [V] (from English peak ).

The information always relates to the resolution bandwidth used in Hertz. For example, a sinusoidal signal with a voltage profile of V at a terminating resistor of 50  ohms generates an effective voltage of 30 dBm or 16.9897 dBV or 7.0711 V (RMS) or 10 V (PK) for each resolution bandwidth. ${\displaystyle f(t)=10\sin(\omega t)}$

Examples

LDS of a monofrequency signal with quantization noise

• If the correlation function is a delta distribution , it is called white noise , in this case is constant. ${\displaystyle S_{XX}(\omega )}$
• For the thermal noise , more accurate for the noise power spectral density, where: N ₀  =  k _B  ·  T . At 27 ° C it is 4 · 10 ⁻²¹  J = 4 · 10 ⁻²¹  W / Hz = −204 dBW / Hz = −174 dBm / Hz
• The picture on the right shows an MSS of the function with a uniformly distributed noise process ( quantization noise ) at a sampling rate of 44,100 Hz and a resolution bandwidth of BW = 43.1 Hz (resulting from 44100 Hz / 1024  FFT points), as it is, for example, from a  CD could come. The peak at about −3 dB represents the sine signal on the noise floor at about −128 dB. Since the power information relating to the resolution bandwidth, can be the SNR to read (note the Logarithmusgesetz , the multiplications in additions transformed). The SNR read off from the picture thus comes very close to the theoretically expected one . ${\displaystyle f(t)=\sin(2\pi 3500t)+2^{-16}R(t)}$ ${\displaystyle |R(t)|\leq 1}$ ${\displaystyle -3-(-128+10\log _{10}(22050/{\text{BW}}))=97{,}9\,\mathrm {dB} }$ ${\displaystyle 10\log _{10}(1^{2}/2)-10\log _{10}((2^{-16})^{2}/3)=98{,}0905\,\mathrm {dB} }$

See also

• Parseval's equation
• Parseval's theorem
• Spectral acceleration density
• Spectral representation of stationary stochastic processes
• Spectral density estimation

literature

• Hans Dieter Lüke: Signal transmission. Basics of digital and analog communication systems . 6th revised and expanded edition. Springer, Berlin et al. 1995, ISBN 3-540-58753-5 .

Individual evidence

1. ^ Karl-Dirk Kammeyer , Kristian Kroschel: Digital signal processing. Filtering and Spectral Analysis. With MATLAB exercises. 6th corrected and supplemented edition. Teubner, Stuttgart et al. 2006, ISBN 3-8351-0072-6 , chap. 8.3, p. 315ff.
2. ^ Karl-Dirk Kammeyer , Kristian Kroschel: Digital signal processing. Filtering and Spectral Analysis. With MATLAB exercises. 6th corrected and supplemented edition. Teubner, Stuttgart et al. 2006, ISBN 3-8351-0072-6 , chap. 8.4, p. 326ff.