The power spectrum (PS) of a timedomain signal is the distribution of power contained within the signal over frequency, based on a finite set of data. The frequencydomain representation of the signal is often easier to analyze than the timedomain representation. Many signal processing applications, such as noise cancellation and system identification, are based on the frequencyspecific modifications of signals. The goal of the power spectral estimation is to estimate the power spectrum of a signal from a sequence of time samples. Depending on what is known about the signal, estimation techniques can involve parametric or nonparametric approaches, and can be based on timedomain or frequencydomain analysis. For example, a common parametric technique involves fitting the observations to an autoregressive model. A common nonparametric technique is the periodogram. The power spectrum is estimated using Fourier transform methods such as the Welch method and the filter bank method. For signals with relatively small length, the filter bank approach produces a spectral estimate with a higher resolution, a more accurate noise floor, and peaks more precise than the Welch method, with low or no spectral leakage. These advantages come at the expense of increased computation and slower tracking. For more details on these methods, see Spectral Analysis. You can also use other techniques such as the maximum entropy method.
In Simulink^{®}, you can perform realtime spectral analysis of a dynamic signal using the
Spectrum Analyzer block. You can view the spectral data in the spectrum
analyzer. To acquire the last spectral data for further processing, create a Spectrum Analyzer Configuration
object and run the getSpectrumData
function on this object. Alternately, you can use the
Spectrum Estimator block from the
dspspect3
library to compute the power spectrum, and Array Plot block to view the spectrum.
You can view the power spectrum (PS) of a signal using the Spectrum Analyzer block. The PS is computed in real time and varies with the input signal, and with changes in the properties of the Spectrum Analyzer block. You can change the dynamics of the input signal and see what effect those changes have on the spectrum of the signal in real time.
The model ex_psd_sa
feeds a noisy sine wave signal to the
Spectrum Analyzer block. The sine wave signal is a sum of two
sinusoids: one at a frequency of 5000 Hz and the other at a frequency of 10,000 Hz.
The noise at the input is Gaussian, with zero mean and a standard deviation of
0.01.
Open and Inspect the Model
To open the model, enter ex_psd_sa
in the MATLAB^{®} command prompt.
Here are the settings of the blocks in the model.
Block  Parameter Changes  Purpose of the block 

Sine Wave 1 

Sinusoid signal with frequency at 5000 Hz 
Sine Wave 2 

Sinusoid signal with frequency at 10000 Hz 
Random Source 
 Random Source block generates a random noise signal with properties specified through the block dialog box 
Add  List of signs to
+++ .  Add block adds random noise to the input signal 
Spectrum Analyzer 
Click the Spectrum Settings icon . A pane appears on the right.
Click the Configuration Properties
icon and set Ylimits
(Minimum) as  Spectrum Analyzer block shows the Power Spectrum Density of the signal 
Play the model. Open the Spectrum Analyzer block to view the power spectrum of the sine wave signal. There are two tones at frequencies 5000 Hz and 10,000 Hz, which correspond to the two frequencies at the input.
RBW, the resolution bandwidth is the minimum frequency
bandwidth that can be resolved by the spectrum analyzer. By default, RBW
(Hz) is set to Auto
. In the
Auto
mode, RBW is the ratio of
the frequency span to 1024. In a twosided spectrum, this value is F_{s}/1024, while in a onesided spectrum, it is (F_{s}/2)/1024. The spectrum analyzer in ex_psd_sa
is
configured to show onesided spectrum. Hence, the RBW is (44100/2)/1024 or 21.53
Hz.
Using this value of RBW, the number of input samples used to compute one spectral update is given by N_{samples} = Fs/RBW, which is 44100/21.53 or 2048.
RBW calculated in this mode gives a good frequency resolution.
To distinguish between two frequencies in the display, the distance between the two frequencies must be at least RBW. In this example, the distance between the two peaks is 5000 Hz, which is greater than RBW. Hence, you can see the peaks distinctly. Change the frequency of the second sine wave from 10000 Hz to 5015 Hz. The difference between the two frequencies is less than RBW.
On zooming, you can see that the peaks are not distinguishable.
To increase the frequency resolution, decrease RBW to 1 Hz and run the simulation.
On zooming, the two peaks, which are 15 Hz apart, are now distinguishable
When you increase the frequency resolution, the time resolution decreases. To
maintain a good balance between the frequency resolution and time resolution, change
the RBW (Hz) to Auto
.
Change the Input Signal
When you change the dynamics of the input signal during simulation, the power
spectrum of the signal also changes in real time. While the simulation is running,
change the Frequency of the Sine Wave 1 block to
8000
and click Apply. The second tone in
the spectral analyzer output shifts to 8000 Hz and you can see the change in real
time.
Change the Spectrum Analyzer Settings
When you change the settings in the Spectrum Analyzer block, the effect can be seen on the spectral data in real time.
When the model is running, in the Trace options pane of the
Spectrum Analyzer block, change the Scale to
Log
. The PS is now displayed on a log scale.
For more information on how the Spectrum Analyzer settings affect the power spectrum data, see the 'Algorithms' section of the Spectrum Analyzer block reference page.
The spectrum analyzer provides six units to specify the power spectral density: Watts/Hz
,
dBm/Hz
, and dBW/Hz
. Corresponding units of power are
Watts
, dBm
, and dBW
. For electrical
engineering applications, you can also view the RMS of your signal in Vrms
or
dBV
. The default spectrum type is Power in
dBm
.
Power in dBm
is given by:
$${P}_{dBm}=10\mathrm{log}10(power\text{\hspace{0.17em}}in\text{\hspace{0.17em}}watt/1\text{\hspace{0.17em}}milliwatt)$$
Power in dBW
is given by:
$${P}_{dBW}=10\mathrm{log}10(power\text{\hspace{0.17em}}in\text{\hspace{0.17em}}watt/1\text{\hspace{0.17em}}watt)$$
For a sine wave signal with an amplitude of 1 V, the power of
a onesided spectrum in Watts
is given
by:
$$\begin{array}{l}{P}_{Watts}={A}^{2}/2\\ {P}_{Watts}=1/2\end{array}$$
$$\begin{array}{l}{P}_{dBm}=10\mathrm{log}10(power\text{\hspace{0.17em}}in\text{\hspace{0.17em}}watt/1\text{\hspace{0.17em}}milliwatt)\\ {P}_{dBm}=10\mathrm{log}10(0.5/{10}^{3})\end{array}$$
For a white noise signal, the spectrum is flat for all frequencies. The spectrum analyzer in this example shows a onesided spectrum in the range [0 Fs/2]. For a white noise signal with a variance of 1e4, the power per unit bandwidth (P_{unitbandwidth}) is 1e4. The total power of white noise in watts over the entire frequency range is given by:
$$\begin{array}{l}{P}_{whitenoise}={P}_{unitbandwidth}*number\text{\hspace{0.17em}}of\text{\hspace{0.17em}}frequency\text{\hspace{0.17em}}bins,\\ {P}_{whitenoise}=({10}^{4})*\left(\frac{Fs/2}{RBW}\right),\\ {P}_{whitenoise}=({10}^{4})*\left(\frac{22050}{21.53}\right)\end{array}$$
If you set the spectral units to dBFS
and set the full scale (FullScaleSource) to Auto
, power in dBFS
is computed as:
$${P}_{dBFS}=20\cdot {\mathrm{log}}_{10}\left(\sqrt{{P}_{watts}}/Full\_Scale\right)$$
P_{watts}
is the power in watts
For double and float signals, Full_Scale is the maximum value of the input signal.
For fixed point or integer signals, Full_Scale is the maximum value that can be represented.
If you specify a manual full scale (set FullScaleSource to
Property
), power in dBFS
is given by:
$${P}_{FS}=20\cdot {\mathrm{log}}_{10}\left(\sqrt{{P}_{watts}}/FS\right)$$
Where FS
is the full scaling factor specified in the FullScale property.
For a sine wave signal with an amplitude of 1 V, the power of a onesided spectrum in
Watts
is given by:
$$\begin{array}{l}{P}_{Watts}={A}^{2}/2\\ {P}_{Watts}=1/2\end{array}$$
$${P}_{FS}=20\cdot {\mathrm{log}}_{10}\left(\sqrt{1/2}/1\right)$$
Fs = 1000; % Sampling frequency sinef = dsp.SineWave('SampleRate',Fs,'SamplesPerFrame',100); scope = dsp.SpectrumAnalyzer('SampleRate',Fs,... 'SpectrumUnits','dBFS','PlotAsTwoSidedSpectrum',false) %% for ii = 1:100000 xsine = sinef(); scope(xsine) end
Power in dBm
is given by:
$${P}_{dBm}=10\mathrm{log}10(power\text{\hspace{0.17em}}in\text{\hspace{0.17em}}watt/1\text{\hspace{0.17em}}milliwatt)$$
Voltage in RMS is given by:
$${V}_{rms}={10}^{{P}_{dBm}/20}\sqrt{{10}^{3}}$$
From the previous example, P_{dBm} equals 26.9897 dBm. The V_{rms} is calculated as
$${V}_{rms}={10}^{26.9897/20}\sqrt{0.001}$$
To confirm this value:
Change Type to RMS
.
Open the peak finder by clicking Tools > Measurements > Peak Finder.
Alternately, you can compute the power spectrum of the signal using the Spectrum Estimator block in the dspspect3
library.
You can acquire the output of the spectrum estimator and store the data for further
processing.
Replace the Spectrum Analyzer block in ex_psd_sa
with the Spectrum Estimator block followed by an
Array Plot block. To view the model,
enter ex_psd_estimatorblock
in the MATLAB command prompt. In addition, to access the spectral estimation data in
MATLAB, connect the To Workspace block to the output of the
Spectrum Estimator block. Here are the changes to the settings of
the Spectrum Estimator block and the Array Plot
block.
Block  Parameter Changes  Purpose of the block 

Spectrum Estimator 
 Computes the power spectrum of the input signal using the filter bank approach. 
Array Plot 
Click View and
 Displays the power spectrum data. 
The spectrum displayed in the Array Plot block is similar to the
spectrum seen in the Spectrum Analyzer block in
ex_psd_sa
.
The filter bank approach produces peaks that have very minimal spectral leakage.
Convert x
axis to Represent
Frequency
By default, the Array Plot block plots the PS data with respect to
the number of samples per frame. The number of points on the xaxis equals the
length of the input frame. The spectrum analyzer plots the PS data with respect to
frequency. For a onesided spectrum, the frequency varies in the range [0 Fs/2]. For
a twosided spectrum, the frequency varies in the range [Fs/2 Fs/2]. To convert the
x
axis of the array plot from samplebased to
frequencybased, do the following:
Click on the Configuration Properties icon . On Main tab, set
Sample increment to
Fs/FrameLength
.
For a onesided spectrum, set Xoffset to
0
.
For a twosided spectrum, set Xoffset to
Fs/2
.
In this example, the spectrum is onesided and hence, the
Sample increment and Xoffset are set
to 44100/1024
and 0
, respectively. To specify
the frequency in kHz
, set the Sample
increment to 44.1/1024
.
Live Processing
The output of the Spectrum Estimator block contains the spectral
data and is available for further processing. The data can be processed in realtime
or it can be stored in the workspace using the To Workspace block.
This example writes the spectral data to the workspace variable
Estimate
.