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Tutorial on Measurement of Power Spectra

National Instruments Inc.

The Fast Fourier Transform (FFT) and the power spectrum are powerful tools for analyzing and

measuring signals from plug-in data acquisition (DAQ) devices. For example, you can effectively

acquire time-domain signals, measure the frequency content, and convert the results to real-world

units and displays as shown on traditional benchtop spectrum and network analyzers. By using

plug-in DAQ devices, you can build a lower cost measurement system and avoid the

communication overhead of working with a stand-alone instrument. Plus, you have the flexibility

of configuring your measurement processing to meet your needs.

To perform FFT-based measurement, however, you must understand the fundamental issues and

computations involved. This application note serves the following purposes.

? Describes some of the basic signal analysis computations,

? Discusses antialiasing and acquisition front ends for FFT-based signal analysis,

? Explains how to use windows correctly,

? Explains some computations performed on the spectrum, and

? Shows you how to use FFT-based functions for network measurement.

The basic functions for FFT-based signal analysis are the FFT, the Power Spectrum, and the

Cross Power Spectrum. Using these functions as building blocks, you can create additional

measurement functions such as frequency response, impulse response, coherence, amplitude

spectrum, and phase spectrum.

FFTs and the Power Spectrum are useful for measuring the frequency content of stationary or

transient signals. FFTs produce the average frequency content of a signal over the entire time that

the signal was acquired. For this reason, you should use FFTs for stationary signal analysis or in

cases where you need only the average energy at each frequency line. To measure frequency

information that is changing over time, use joint time-frequency functions such as the Gabor

Spectrogram.

This application note also describes other issues critical to FFT-based measurement, such as the

characteristics of the signal acquisition front end, the necessity of using windows, the effect of

using windows on the measurement, and measuring noise versus discrete frequency components.

Basic Signal Analysis Computations

The basic computations for analyzing signals include converting from a two-sided power

spectrum to a single-sided power spectrum, adjusting frequency resolution and graphing the

spectrum, using the FFT, and converting power and amplitude into logarithmic units.

The power spectrum returns an array that contains the two-sided power spectrum of a time-

domain signal. The array values are proportional to the amplitude squared of each frequency

component making up the time-domain signal. A plot of the two-sided power spectrum shows

negative and positive frequency components at a height

where Ak is the peak amplitude of the sinusoidal component at frequency k. The DC component

has a height of A0

2 where A0 is the amplitude of the DC component in the signal.

Figure 1 shows the power spectrum result from a time-domain signal that consists of a 3 Vrms

sine wave at 128 Hz, a 3 Vrms sine wave at 256 Hz, and a DC component of 2 VDC. A 3 Vrms

sine wave has a peak voltage of 3.0 or about 4.2426 V. The power spectrum is computed

from the basic FFT function. Refer to the Computations Using the FFT section later in this

application note for an example this formula.

Figure 1. Two-Sided Power Spectrum of Signal

Converting from a Two-Sided Power Spectrum to a Single-Sided Power Spectrum

Most real-world frequency analysis instruments display only the positive half of the frequency

spectrum because the spectrum of a real-world signal is symmetrical around DC. Thus, the

negative frequency information is redundant. The two-sided results from the analysis functions

include the positive half of the spectrum followed by the negative half of the spectrum, as shown

in Figure 1.

In a two-sided spectrum, half the energy is displayed at the positive frequency, and half the

energy is displayed at the negative frequency. Therefore, to convert from a two-sided spectrum to

a single-sided spectrum, discard the second half of the array and multiply every point except for

DC by two.

where SAA(i) is the two-sided power spectrum, GAA(i) is the single-sided power spectrum, and N

is the length of the two-sided power spectrum. The remainder of the two-sided power spectrum

SAA

The non-DC values in the single-sided spectrum are then at a height of

This is equivalent to

where

is the root mean square (rms) amplitude of the sinusoidal component at frequency k. Thus, the

units of a power spectrum are often referred to as quantity squared rms, where quantity is the unit

of the time-domain signal. For example, the single-sided power spectrum of a voltage waveform

is in volts rms squared.

Figure 2 shows the single-sided spectrum of the signal whose two-sided spectrum Figure 1 shows.

Figure 2. Single-Sided Power Spectrum of Signal in Figure 1

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Title: Tutorial on Measurement of Power Spectra

Author: Tony Tyson

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