Weigh the Alternatives for Spectral Analysis
Spectrum analyzers, FFT analyzers, vector analyzers, and DSOs can all measure frequency spectrums. Your application will determine which instrument you should use.
Allen Montijo, Agilent Technologies, Colorado Springs, CO -- Test & Measurement World, 11/1/1999
Signal characteristics that are difficult to see in the time domain frequently become visible in the frequency domain. The classic example is a square wave, which appears as the sum of many harmonics in the frequency domain. When you want to see the spectral content of signals, you have several choices of test equipment— spectrum analyzers, Fourier analyzers, vector signal analyzers, and digitizing oscilloscopes. Each has advantages and disadvantages, and your choice will depend on your application (See Table 1). Spectrum analyzers use a sweeping signal source, a superheterodyne mixer, and filters to sweep across the measurement range and measure the power in one frequency band at a time (Fig. 1). In effect, a spectrum analyzer is a bank of narrowband filters, each tuned to a different frequency at regular intervals. Spectrum analyzers have the widest frequency range of all the instruments. They can analyze signals from tens of hertz into the hundreds of gigahertz over a wide dynamic range.
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Figure 1. A spectrum analyzer sweeps across a frequency range, effectively applying a series of narrowband filters to a CW signal. |
If you measure continuous wave (CW) signals, with or without modulation, then a spectrum analyzer is your ideal instrument. EMI engineers also use these instruments to perform precompliance EMI scans.
Because they measure only one frequency band at a time, spectrum analyzers have three main drawbacks. First, they can measure only steady-state signals, so they are impractical for measuring transient events. Second, because spectrum analyzers can measure only one frequency at a time, you lose all information about the relative phases of the frequency components. The instrument displays power magnitude only—a scalar quantity. Third, measurement times are often a factor of 10 to 100 times longer than with fast Fourier transform (FFT) techniques.
FFTs Are Like Filters
Fourier analyzers—sometimes called FFT analyzers—add complex (vector) analysis to your measurements. These analyzers use 12-bit, 14-bit, or 16-bit ADCs to digitize a waveform, and they store as many as several million samples. Then, they apply an FFT algorithm to translate the data into the frequency domain.
Performing the FFT is analogous to applying a bank of narrowband filters to the digitized signal all at once (Fig. 2). Because the FFT analyzer digitizes a signal, it can apply the FFT algorithm to all of the signal’s frequencies at the same time. The FFT algorithm does its work after the signal acquisition rather than in "real time" like a spectrum analyzer. So FFT analyzers avoid having to sequentially apply a series of filters to your signal the way spectrum analyzers do.
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| Figure 2. An FFT analyzer samples a signal and calculates its frequency content. |
The demands of good linearity and high resolution, however, limit the ADCs used in FFT analyzers to those with relatively low sample rates—from a few hundred ksamples per second to several Msamples/s. This in turn limits the frequency range of these instruments to about 100 kHz, with some capable of a few megahertz. Deep memory allows these instruments to have frequency resolutions of better than 1 mHz. Because of the FFT analyzer’s low frequency resolution, you can use FFT analyzers for analysis of vibration and acoustics. You also can use them when testing low-frequency electronics (including control systems) and low-frequency communications.
FFT analysis requires only the time for the filters to settle once (the data acquisition), plus the FFT computation time. Furthermore, these times usually get overlapped so that only the longer time limits the instrument’s throughput. FFT analyzers specify a real-time frequency range, below which the analyzer is continuously monitoring the input signals—all of your data contributes to all of your measurements. In contrast, if you’re using a spectrum analyzer to measure N frequency bins, the filter must settle N times, and a specific time-slice of your data can affect only the frequency bin that the analyzer was capturing at the time.
FFT analysis also lets you see the relative phases of a signal’s spectral components. When you measure a transient event such as an impulse, you can use the instrument to calculate the complex transfer function of your circuit. Some analyzers will even calculate the transfer function’s poles and zeros for you. Phase information also lets you display the data in more useful formats, such as a Nyquist plot or a group-delay plot.
Because FFT analyzers digitize and store signals, they also can show you how a signal’s frequency content changes over time. FFT analyzers can give you that third dimension (time) by generating a waterfall display (Fig. 3), which shows the results of many acquisitions at once. Each trace gets offset slightly from the previous one, generating a 3-D-ish image that often resembles a waterfall.
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| Figure 3. A waterfall display shows the frequency content of many signal acquisitions so you can view how a signal’s frequency content changes over time. |
FFTs at Higher Frequencies
FFT analyzers are no good for RF signals. Spectrum analyzers can measure RF signals, but they lose phase information. When you need FFT analysis for RF signals, you can use a vector analyzer. The vector signal analyzer extends the FFT analyzer’s capabilities to higher frequencies for narrowband signal analysis. These instruments are ideal for analyzing communications signals. Their capabilities include digital modulation analysis and AM/ FM/PM demodulation.
A vector analyzer adds a spectrum analyzer front end to an FFT analyzer (Fig. 4). The spectrum-analyzer front end extends the instrument’s capabilities into the gigahertz range. As with the spectrum analyzer, a mixer and filter shift the input signal in frequency. In this case, the analyzer shifts the desired input signal’s frequency range down to a range compatible with the back-end Fourier analyzer. The filter does not have to be as selective as it is for the spectrum analyzer because its purpose is to prevent aliasing and not to measure the spectral lines. The filter’s frequency range can be almost as large as the input frequency range of the Fourier analyzer. As a result, you can use a vector analyzer to view both continuous and transient narrowband RF signals in the frequency domain.
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Figure 4. A vector signal analyzer combines the features of a spectrum analyzer with those of an FFT analyzer to give you FFT analysis at RF frequencies. |
DSOs Catch Transient Events
Like vector analyzers, digitizing oscilloscopes (DSOs) extend the FFT technique to higher frequencies than those possible with Fourier analyzers. Many DSOs provide you with basic spectral analysis as a math function. But DSOs don’t have the 12-bit or 16-bit amplitude resolution of FFT analyzers. High-speed DSOs typically resolve signals to 8 bits. Still, their high sampling rate (several billion samples per second) lets you see the spectrum of transient events such as ESD.
Be cautious when you use a DSO’s frequency-domain capabilities, because DSOs are optimized for time-domain analysis. Small time-domain distortions introduced by the DSO’s front end that are difficult to see in the time domain become obvious in the frequency domain, especially when you view a signal’s amplitude in decibels. These distortions introduce low-level harmonics into your scope’s FFT when you’re viewing simple signals such as sine waves.
In Figure 5, a square wave’s harmonics decrease at roughly a linear rate (in decibels) from left to right. These harmonics are numbered 2, 3, 11, 12, and 13. The right border of the display represents the fold-back frequency (one-half of the sample rate). Aliased harmonics (14 and 15) beyond the fold-back frequency reflect, or fold, about this border and appear as decreasing amplitudes from right to left. As long as the spectral lines are distinct, you can make accurate amplitude measurements on any of the harmonics, even the aliased ones. If you know how many times a harmonic has "reflected" or "folded" from the edges of the display, then you can correct the measured frequency (DC to the fold-back frequency, FS/2) to find the actual frequency. When aliasing causes harmonics to overlap, you can’t measure the one with the smaller amplitude. Your measurement accuracy on the larger harmonic depends on the harmonics’ relative amplitudes; you’re measuring the result of a vector sum.
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Figure 5. Above the 13th harmonic, all other harmonics of the square wave are aliased and “reflect” off the right side of the display. Harmonics above the 26th alias again and reflect back from left to right. |
Many high-sample-rate DSOs achieve their high sample rates by interleaving several fast ADCs per channel. The ADCs vary slightly in linearity, gain, offset, frequency response, and timing. These mismatches, interleaved in time, introduce extraneous signals at multiples of FS/N into your scope, where N is the number of interleaved ADCs and FS is the interleaved sample rate. Except for offset, all of these errors result in mixing of your signal and the FS/N components. For example, assume that four 1-Gsample/s ADCs are interleaved to provide 4 Gsamples/s. The FS/N components are at 1 GHz and 2 GHz. Offset differences between the ADCs result in spectral components at 1 GHz and 2 GHz. With a 100-MHz sine-wave input, timing errors result in fictitious spectral components at 900, 1100, 1900.
High-speed scopes have 8 bits of resolution as opposed to the 12 bits or more in FFT analyzers. Fewer bits result in more quantization errors. The noise power of an ideal ADC decreases by 6.02 dB for each additional bit of resolution. With fewer bits than FFT analyzers and vector signal analyzers, your DSO’s quantization noise can add nonlinearities to a signal. The DSO’s noise floor is higher, by 20 to 40 dB, and the nonlinearities of the ADC process are correspondingly higher than in an FFT analyzer. Nonlinearities contaminate the FFT by allowing the quantization noise to mix with your signal, generating spurious spectral components instead of random noise. Fortunately, if your signal contains "reasonable" slew rates (greater than 1 quantization level per sample), this quantization noise tends to become random, spreading the power uniformly across the spectrum. Many DSOs add a controlled "noise" source to your signal before quantization and subtract it after quantization. This "dithering" is designed to optimize the frequency content of the signal and minimize the effects of nonlinearities in the ADC, including the quantization process.
Whenever you can, use the DSO’s real-time capture mode—as opposed to equivalent-time sampling mode—for spectrum analysis. The FFT assumes that the samples are equally spaced, so just a few picoseconds of sampling jitter can destroy the FFT’s signal-to-noise ratio(SNR). The samples from an equivalent-time record represent many independent acquisitions. Timing inconsistencies in your circuit from repetition to repetition and DSO trigger jitter easily result in tens of picoseconds of effective jitter, making the FFT useless. If you must use an equivalent-time mode, set the number of averages, N, as high as possible. This will statistically reduce your jitter by the square-root of N.
Even with these issues, your DSO’s FFT lets you gain insight into the frequency-domain behavior of your circuit. The high sample rate of DSOs lets you view the spectrum of high-speed transients as well as continuous signals. T&MW
Allen Montijo is an engineer/scientist in the Electronic Product Solutions Group of Agilent Technologies, where he designs hardware and software for digital oscilloscopes. He holds a B.S.E. in electrical engineering/computer science from Princeton University and an M.S.E.E. from Stanford University. E-mail: montijo@col.hp.com.
