frequency spectrum

The representation of a signal in the frequency domain. The signal is broken into multiple periodic signals, each with an amplitude and phase. The frequency spectrum is useful for identifying repeating signals, of which a sine wave is the simplest.

Types of Frequency Spectra

A wide range of terminology has been used to describe various types of frequency spectra. Frequency spectra can be broken down into three main groups:


1. Frequency Spectrum. (also known as Instantaneous Spectrum, Fourier Spectrum, FFT Spectrum, or Complex Spectrum)

Here, the spectrum is measured on one block of data and no averaging is performed. The spectrum consists of a number of periodic components (one vector per frequency), which are most often displayed as magnitude and phase information. Most instruments do not show the phase part. Normally the magnitude is expressed in Volts or power ( V²). When the FFT is used the compute the Frequency Spectrum, the frequency components will have a linear spacing. For example, a 1000 point time block will be transformed into 500 frequency components which are equally spaced. If constant percentage bandwidth analysis is used (such as for 1/3 octave analysis), then the frequency components are spaced on a logarithmic axis. Each filter has a bandwidth which is a given percentage (26% for 1/3 octave) of the center frequency of the filter.

2. Averaged Complex Spectrum. (also known as Averaged Fourier Spectrum)

Since a complex spectrum consists of both a magnitude and phase, it is only meaningful to performing averaging of these spectra for repeated signals acquired under identical trigger conditions. This results in the acquired waveform being positioned with the same delay relative to the trigger. The averaging is done on the real and imaginary components individually, so that the phase information is retained. This type of measurement picks out periodic frequency components and is very useful in removing noise that is not correlated to the repeated signal.

If complex averaging is performed on non-repeating signals, the result will converge toward zero, since the phases of the individual frequency components will be randomly distributed for each acquisition and spectrum calculation.

To see this in practice, run demo1 using the LabVIEW Player. Set the Number of Averages to 4000, the Weighting Mode to "Linear", and set the Trigger "On". This sets up the instrument to perform synchronous averaging on the sine signal. Now switch the Averaging Mode between "RMS Averaging" and "Vector Averaging". Notice how the vector averaging reduces the noise, without affecting the level of the sine wave. Now turn the trigger off, while still using "Vector Averaging" and notice that the level of the sine wave now jumps around, since it is no longer being synchronously averaged.

To see the effect of synchronous averaging on the time domain signal, run demo 2 where you can also find a sine wave buried in noise.


3. Power Spectrum. (also known as autospectrum, autopower spectrum, or Spectrum)

These terms refer to an average of the power of the individual frequency components over a number of instantaneous spectra. When this averaging occurs, the magnitude of each frequency component is squared, and added to the previous sum for that frequency. Hence, the phase information is discarded.

The averaging to compute a power spectrum does not reduce the unwanted noise in the system. However, it can be extremely useful in getting a reliable statistical estimate of the level of random signals being measured. For example, if you only display the instantaneous spectrum of white noise, the spectrum will be extremely jagged. But after computing the power spectrum with adequate averaging, the shape of the spectrum will converge on a flat spectrum. Power spectra can be shown in units of power, or the square root of the averaged result can be taken to give a voltage value (which is the RMS value).

In addition, the power spectrum may be scaled in several different ways, depending on the type of signal you are analyzing:: Power Spectral Density (PSD) and Energy Spectral Density (ESD).

• PSD is used when measuring continuous broadband noise, and normalizes the power to an equivalent bandwidth of 1 Hz, irrespective of the actual bandwidth of the filter being used. For example, if a signal is measured at -93 dB in a 10 Hz bandwidth, then the spectral density would be -103 dB (in a 1 Hz bandwidth). This makes it possible to compare noise measurements made with different bandwidth settings of the spectrum analyzer.

• ESD is used to measure the energy of transient signals. Since transients also are spread out over a broad frequency range, they must be normalized to 1 Hz (as with noise). In addition, the duration of transients may vary significantly, so their duration is also normalized to a standard equivalent duration of 1 second. This makes it possible to compare the spectra of different transients.

Notes on Terminology:

The term "frequency response of a signal" is often loosely used, but incorrectly. Frequency response refers to the transfer characteristic of a system, that is, the input/output relationship. For example, the gain and phase response of a filter. Likewise, the term "the spectrum of a filter" is often incorrectly used when what really is mean is the frequency response of the filter.

Example of Measurements of Frequency Spectra on Random Noise Signals

If you are making an RMS measurement on a random noise signal, the error of the measurement will typically be dominated by the effective number of statistical averages you perform, and not as much on the actual "sine wave" (data sheet) accuracy of the instrument. The random error (with 95% confidence which corresponds to 2 sigma (standard deviations)) of a measurement on random noise signals is equal to


where B = Bandwidth of the measurement in Hz
and T = the averaging time in seconds.

For example, if you measure random noise with a bandwidth of 100 Hz over 1 second, the error will be equal to

This means the correct value will lie between 0.95 and 1.05, which is within approximately 0.5 dB.

This equation is solved interactively and graphically in this link.

Demo 3 illustrates how the averaging reduces the variation of the correct value of the spectral componenets of random noise. When you run the demo, notice that as the number of averages increases, the noise floor of the spectrum becomes smoother, as it converges to the correct value as determined by the equation above.

If you are measuring a sine wave and there is significant random noise present, the noise will bias the value of the sine wave measurement to be too high. In cases of this nature, it is best to use a spectrum analysis technique where the sine wave component can be separated from the random noise components. The longer the measurement, the more noise can be removed, and the less the value of the sine wave will be affected by the noise. Long measurement times make very narrow spectrum measurements possible. This is also illustrated in demo 3 using the LabVIEW Player.

Units:

V vs Hz
Additional References

Related NI Products:

• Sound and Vibration Toolset
  
  Helpful Web Sites:
  
• FFT and Spectrum Analysis in LabVIEW
  
• FFT and Signal Analysis Tutorial
  
• Advanced interactive demonstration of frequency spectrum measurements
  
  

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