Below are the slides I had prepared for a talk about FFT analysis that may help you understand some of the topics related to the psdZoom application.

# Some practical aspects of data acquisition Speaker: Paul Mennen
Sampled data systems Anti-alias filtering A/D conversion Averaging Windowing Frequency translation Signal generation FFT Analyzers

First I'll talk about sampled data systems in general, then more detail on the remaining topics. At the end, I'll talk about the analyzers that employ these ideas. Since I have built and supported these analyzers, I've been exposed to many of the practical concerns and problems that are not often found in books. I would like to share some of those with you today.
A sampled data system includes the acquisition section or the signal generation section or both.
Acquisition

• | Signal Conditioning: Preamps, bias, AA filters, etc.
• | A/D: Discretize signal in both time & amplitude
• | Signal Processing: DFT and more

Generation

• | Signal Processing: data from external source or from acquisition
• | D/A: Convert signal to analog form (zero order hold)
• | Signal Conditioning: Smoothing filters, buffers, power amps, etc

I've heard people say that they will be able to tell an alias when they see one, so it's not worth the expense to have AA filters.
Of course, it is not possible to tell. But if you are not convinced, consider this transfer function of an electrical network:

• | Blue trace: with AA filters (accurate). Substantial notch at 3.3kHz.
• | Red trace: without AA filters. Not only is the notch not nearly as deep, but it shifted frequency as well. The error at the high end is nearly 20dB.

Obviously, without AA filters, you would have no way to know that this transfer function estimate was bogus. Often the lack of AA filtering will cause a jagged appearance to the transfer function and/or low coherence. Not always, however. In this case, averaging smoothed out the erroneous estimate, and the coherence was near 1.
Note: A transfer function is the ratio of the power of the system output to the power of the system excitation measured at each frequency.
Here is an example of a typical analog AA filter for use with a conventional A/D converter.
These are the important characteristics of AA filters.

• | Passband cutoff frequency:This parameter defines the bandwidth of the analysis. In this example, we have chosen a bandwidth of 20kHz. This means that we will be able to analyze signals in the DC to 20kHz range. (This covers most audio and vibration applications.)
• | Passband ripple: This example shows several dB of ripple, which is larger than you would normally want in a high-quality front end. The ripple in the passband degrades the amplitude accuracy spec. To reduce this ripple (all other things being equal) we would need to use a larger order filter.
• | Stopband rejection: About -80dB in this example. More is better. Dynamic range can never be better than the amount of stopband rejection.
• | Efficiency: The 3rd and last important parameter relates to how steeply the response transitions from the pass band to the stop band. Usually specified in terms of roll-off in dB/octave or dB/decade. A more useful way of specifying this parameter I will call "efficiency".

Since we decided that we want to analyze signals up to 20 kHz, the Nyquist theorem tells us the sample rate must be 40 kHz or higher. Suppose we choose a sample rate of 40 kHz. Now imagine an input signal consisting solely of a 20.1 kHz sine wave. When sampled at 40 kHz, the resulting samples will look the same as if the input came from a 0.1 kHz sine wave - i.e. an "alias". (Convince yourself of this by trying it.) So to prevent this alias (and others) the analog anti-aliasing filter would need a stop band that starts at 20kHz. But the filter should be flat in the passband (up to 20kHz), so the filter response would have to go straight down from one to zero at 20kHz (referred to as "a brick wall filter"). It is impossible to build a brick wall filter, which means we actually have to choose a sample rate that is higher than 40 kHz. As we increase the sample rate, designing and manufacturing the AA filter gets easier, but we also don't want to go higher than necessary since that increases the processing time required for the same analysis.
In this example, the sample rate was chosen to be 51.2 kHz.
You might wonder why I didn't choose some nice round number such as 50 kHz for example. The reason has to do with the frame size which is almost always chosen to be a power of two (since it simplifies the FFT). The frequency resolution (aka FFT bin width) is the sample rate divided by the frame size. Suppose we had chosen 50 kHz for our sample rate and 512 (29) as our frame size. Then the bin width would be 97.65625 Hz. This would work of course, but 100 Hz would be much more convenient (the bin width with a 51.2 kHz sample rate).
Now consider an input signal consisting of a 31.1 kHz sine wave. When sampled at 51.2 kHz, this produces an alias at 20.1 kHz (51.2 - 31.1). However, this is not a problem since we have decided that we are only trying to analyze signals up to 20 kHz. From this calculation, it is apparent that the stop band of the analog AA filter must begin at 31.2 kHz so that no aliases appear anywhere in our 20 kHz bandwidth.
With the filter response shown here, we see that the stop band (above 31.2 kHz) always has an attenuation of 80dB or more, which tells us that any measurement that shows a signal within the 20kHz bandwidth that is above 80dB must be a real signal, and not an alias of some higher frequency input.
The efficiency of the AA filter is the ratio of the analysis bandwidth (20 kHz) over the Nyquist rate (25.6 kHz) which in this case is 78.125%. With higher efficiencies, we get to use a larger fraction of the Nyquist range but the roll-off between the passband and stopband must be steeper (i.e the transition band must be narrower).
AA filtering has its drawbacks as well. If you are making purely time domain measurements, such as estimating time delay or overshoot, then the AA filter makes the job more difficult.
Here is the step response of the AA filter shown previously. The filter overshoot would make it difficult to estimate the overshoot of the system being measured.
For reference, the blue line is the passband magnitude response of the AA filter. The green line is the filter's phase response. Notice that it is not a straight line. This non-linear phase, which is characteristic of efficient AA filters also distort the time history of the signal being measured (but does not affect spectral measurements).
For these reasons, many dynamic signal analyzers provide the ability to bypass the AA-filtering. But when you do, beware. As many of you know from using digital oscilloscopes, occasionally you may completely misinterpret the display as an unsuspected alias comes home to roost.
One problem for an instrument designer is that the AA filter passband cutoff frequency must change depending on the bandwidth required for each specific measurement.
Traditionally this was solved by building the filter with components that can be switched in value using relays or FET switches. For high-performance AA filters this is a difficult design job and an even more difficult manufacturing problem.
Another approach is called a switched capacitor filter, where the passband frequency can be adjusted by controlling a clock rate. However, SCFs are usually limited to lower-quality sample data systems because of their high noise and modest achievable levels of stop-band rejection.
The more modern approach, used when high performance is required, is called a fixed sample rate system. An analog AA filter is used with a fixed cutoff frequency corresponding to the highest bandwidth of interest. If lower sample rates are required, digital decimating low pass AA filters are used to reduce the sample rate. Usually, this filtering is designed using multiple stages of decimate by 2 and decimate by 5 filters. This is more efficient computationally than computing the entire filter in a single stage. Each filter down the chain gets easier because the sample rates get lower and lower.
We could simplify this design noticeably by cascading only the decimate by 2 filters. So instead of the bandwidths shown in this figure, we would get bandwidths of 20kHz, 10kHz, 5kHz, 2.5kHz 1.25kHz, 625Hz, 312.5Hz, 156.25Hz, 78.125Hz, 39.0625Hz, 19.53125Hz, etc. This doesn't prevent you from making all the same measurements, but most users prefer the "nicer" bandwidth and bin width values, justifying the extra design effort for including the decimate by 5 digital filters.
Moving one step down the chain: A/D conversion.
This leads to an important spec of data acquisition systems: Dynamic range
Assume we connect up an ultra-pure sine wave to the input. Theoretically, the power spectrum should show only one peak. Actually, we will see other spurious signals as well. The size of these spurs determines the dynamic range of the acquisition system.
There is no one commonly accepted definition of dynamic range, but this is the one I find most useful.
Quantization noise is a measure of the amplitude error introduced by the A/D because its output can take on only a finite number of values.
Suppose the A/D has b bits. Then the A/D can take on 2b different values. Thus the step size and the quantization noise (in terms of its standard deviation) is proportional to 2-b
So if the noise is proportional to 2-b, the SNR is proportional to 2b.
Taking the log of both sides we get: SNR = about 6 dB per bit plus this constant C to account for the proportionality.
Most books take many pages to derive this formula. Most of the complications have to do with figuring out what the constant C is. However, this constant is often ignored since it is negligible once we are up to a handful of bits. Also, different answers are obtained depending on the assumptions made.
This equation is often interpreted as meaning that the acquisition dynamic range is simply 6 dB times the number of A/D bits. Wrong! The dynamic range can be much worse because of other errors in the system. Also, the dynamic range can be much better since the averaging effects of the FFT and the digital filters reduce the effects of the quantization noise.
So if quantization noise is not the primary limit to dynamic range, what is? The two most likely limitations to the dynamic range are the A/D linearity and the linearity of the signal conditioning circuits.
A/D linearity: The curve in black is for an ideal 4-bit converter. The curve in red shows a realizable converter (exaggerated). Each step of the ideal transfer function is exactly the same size. Such an ideal converter (if you could build such a thing) would pose no limitation to dynamic range. The waviness of the red curve represents the nonlinear behavior. Note in this example there are also two missing codes.
The nonlinearities of the pre-amplifiers or analog AA filters are just as likely to limit the dynamic range. The dynamic range of the measurement can't be considered greater than the amount of rejection in the AA-filter stopband.
Noise from the power supply or digital circuits can leak into the signal conditioning circuits causing spurs that limit dynamic range. Similarly, noise terms can come from external sources such as a CRT monitor sitting on top of the equipment.
The FFT assumes that the signal is uniformly sampled. Clock jitter (usually a problem only with external sampling) violates this assumption and has similar effects to the other forms of non-linearities.
With modern sample and hold circuits, the A/D aperture error is usually small enough not to be a factor.
Moving down the chain again: Averaging.
There are two types of averaging used in a dynamic signal analyzer: Time and frequency domain.
Here is an example of time domain averaging. These five traces are instantaneous (i.e. not averaged) time histories of a repeating signal. When we add them up point-by-point (i.e. column-wise) the result is seen to be much smoother.
For this to work, we must have a stable trigger so that the picture doesn't jump around in the time axis. Without a trigger, the time domain average (the signal and the noise) will eventually average out to zero.
For frequency domain averaging, the idea is similar except that for each time history, we first compute an instantaneous PSD. These five traces are instantaneous PSDs from 5 consecutive frames. Adding them up column-wise as before yields the averaged trace below. Note that the variance of the average result is much less than the instantaneous PSDs. (I actually used 50 averages to get this degree of variance reduction.)
Triggering is not required. Note from the look of the time history (shown in red), that a stable trigger would be impossible with this type of data.
Here is an example that starkly shows the contrast between the two averaging types.
The left side is a PSD computed using frequency domain averaging. It shows a single tone buried in a high level of noise.
The right side is a PSD computed from the result of a time domain average (on the same signal as before). Note that the noise level is much lower since it has been averaged out. This reveals the existence of another tone in the signal (which is the 3rd harmonic). Remember this only works if you have a stable trigger. Also, signal components that are not harmonically related to the fundamental may also average out to zero since they are not synchronously related to the trigger event.
So, more averaging in the frequency domain does not tend to reduce the noise level. It just measures the amount of noise more accurately. However with time domain averaging, the more averaging, the lower the noise.
For both time and frequency averaging types, I have been implicitly talking about additive averaging. That is, as shown in this formula, we add the ith PSD, as i goes from 1 to n (the number of averages) and then divide by n to normalize the result. The second equation is a recursive way of computing the same thing, with the advantage being that the result of each step is correctly normalized so we can view the average as it is being computed. Additive averaging is primarily useful for stationary signals, i.e. signals that don't change their frequency or amplitude characteristics during the time we are viewing them.
If the signal is non-stationary, then additive averaging is not very useful. However, if we still need some degree of variance reduction, we can use exponential averaging to accomplish this while still being able to track changes in the input signal. In this formula, λ (from 0 to 1) controls the tradeoff between the variance reduction provided and the ability to track changes in the input. As λ approaches 0, the old data is weighted less, and the tracking ability approaches that of an instantaneous (non-averaged) measurement. As λ approaches 1, the old data is weighted more resulting in better variance reduction and a longer time constant. [note: the time constant is 1/(1-λ) frames]
Peak hold does not really fit the conventional idea of averaging, but it is convenient to classify it here. Basically, the maximum of the input and previous average is saved, with this determination made separately at each frequency bin. This is most useful when you are trying to determine whether a certain event has occurred over a long measurement time.
A time history is shown at the upper left. The blue trace on the plot to the right shows the magnitude of the DFT of this time history. The green trace on the same plot is actually a more accurate representation of the signal's frequency content. The tone near 16 kHz and the other lumps in the PSD are masked in the DFT by a problem referred to as leakage. The reason is that the blue trace represents the frequency content of the time history repeated an infinite number of times (the periodicity assumptions). Note that the time history begins at a negative voltage but ends at a positive voltage. So when the frame is repeated and placed immediately following, there is a sharp transition from this positive value to the negative one. This sharp transition spreads (or leaks) energy into every bin of the DFT, and obscures the true nature of the frequency content of the signal if we could have sampled it for longer.
Windowing helps solve this problem. Below the time history is the shape of perhaps the most popular window, called Hanning. It is simply a raised cosine function. It is one in the middle and tapers to zero at both ends. If we multiply the time history by this Hanning window, we get the signal in green. Now when we repeat this new signal, no transitions occur at the boundary. The green frequency trace is the magnitude of the DFT of this windowed signal.
One problem with windowing is that it reduces the total energy in the signal which, as we will see, is difficult to account for.
To illustrate one of the drawbacks of windowing, I fabricated a signal consisting of a 1.5 kHz sine wave (at -20dBV) superimposed with a narrow band noise source. The PSD is computed using a boxcar window (i.e. no windowing) and displayed on the left. Notice that the sine wave and the noise source show up at the same amplitude (-20dBV).
Now we apply the Hanning window (in the frequency domain this can be implemented as a convolution with [-.25 .5 -.25]). The result is shown in the middle figure. Now that the leakage has been reduced by the windowing, we can see that the signal also contains a small 1.1kHz tone. (With the boxcar window, that tone was buried by leakage). The power loss due to this window is 3/8 (sum of the squares of the kernel), so we have corrected the plot for this power loss by multiplying by 8/3. Notice that the amplitude of the noise source is the same as before. However, when we cursor the peak at the sine wave, we see an amplitude of -21.7 dB. (1.7 dB smaller than before.)
So, what if we wanted to measure the amplitudes of sinusoidal data? Well, then we can apply the amplitude correction factor instead (the reciprocal of the square of the center term = 4). With this correction factor, we see the display on the right. Now when we cursor the sine wave peak we get the correct amplitude of -20 dBV. But then the amplitude of the noise reads -18.3 dB (again a 1.7 dB error).
One correction factor cannot make both numbers right. The windowing has distorted the data in a non-linear way causing the DFT to change shape. (Explain intuitively by showing how Hanning spreads the power of the sine wave out into the neighboring bins.) This same effect is different for the random portion since the neighboring bins are already at roughly the same level.
The plot on the left is a PSD of a signal containing 2 sine waves separated by 75 Hz. This is 7.5 FFT lines since the bin width is 10Hz.
If we had used the boxcar window (i.e. no windowing) the peak would have been much broader than even the purple trace. So the Hanning window does provide considerable leakage protection but not enough to detect the smaller sine wave. The Blackman-Harris window easily shows the second sine wave. (Blackman-Harris is actually a family of windows, this one known as the minimum 4-term Blackman-Harris).
The plot on the right is the Fourier transform of the window shape. From this, we can see how much leakage protection we get from these two windows. The smaller the side lobes, the smaller the leakage. For example, on the Hanning window curve, at 7.5 bins we see that the side lobe is at about -70dB. But this is the same amplitude as the smaller sine wave, so it is still masked by the leakage. The Blackman-Harris window on the other hand shows about 100 dB of leakage suppression at that position.
The price you pay for the smaller side lobes is the wider main lobe which decreases the frequency resolution of the measurement. (This is the other main drawback of windowing in general. The boxcar window provides the best frequency resolution in cases where the leakage doesn't obscure the measurement.)
One difficulty we encounter with FFT analysis is known as scalloping loss. This is not an easy phenomenon to explain without a lot of heavy math, so in an attempt to make the problem easier to visualize, I fabricated a signal consisting of the sum of 11 equal amplitude sine waves at the frequencies listed here (from 100 Hz up to 1710 Hz). If we sample 256 points of this signal at a 5120 Hz sample rate, our FFT line spacing is Δf = 20 Hz. So dividing by Δf we get the bin numbers shown here. The important part is the fractional component shown in blue. Note that the first component, 100 Hz is exactly on a line (bin 5) and the last component, 1710 is exactly between two lines (bin 85.5), and all the components in between slowly shift from these two extremes as can be seen by looking at the fractional part of the bin number.
Now the PSD (no windowing) is displayed in the upper left. The high leakage is especially noticeable near 2 kHz. Otherwise, the display looks like a good representation of the signal, with one peak for each sine wave component. However, if we expand the amplitude scale (near -10 dB) we see that the amplitudes are not the same for all components as you might expect. The first term (100Hz) is actually correct. The others are in error (up to 3.9 dB). This is the error we refer to as scalloping loss.
When we repeat the experiment with the Hanning window, we see that the leakage is much reduced, but the scalloping loss is still significant (a bit less at 1.4 dB).
Why do we get this scalloping loss? The answer is easily seen from the Fourier transform of the window shapes (shown in the leftmost plot for Boxcar and Hanning). The red dotted lines show the extent of the center bin. The amount of droop within these dotted lines represents the scalloping loss. You can see that Hanning shows less droop than the boxcar.
Now in the center plot, let's look at the transform of two windows that don't have an appreciable scalloping loss. The first one (blue) I'll call the Flat201 window, although it is sometimes called the Potter 201 or the Potter 3-term Flattop window. The second one (green) I'll call the Flat401, and likewise is also known as the Potter 401 or the Potter 5-term Flattop window. Note: Ron Potter (1932-2020) was an engineer at HP who pioneered HP's early development of FFT analyzers.
At the right, we have expanded the center bin so we can measure the scalloping loss. For the Flat201 window the scalloping loss is .018 dB, but by choosing the amplitude correction factor to be in the middle of this range we can say that the error due to scalloping loss is +/- .009 dB. And likewise, for the Flat401, the scalloping loss error is +/- .0033 dB. For either window, these errors are small enough that we can essentially claim victory over the scalloping loss.
But there are still reasons to choose one or the other of these flattop windows (not to mention many other flattop window variations). Notice that Flat401 has more than 80dB of leakage suppression whereas the Flat201 only has about 40dB. So Flat401 is superior when we need this level of leakage suppression. However, this comes at the cost of a much wider main center lobe which reduces the frequency resolution of the measurement. (As you might expect, the Flat301 is a compromise about midway between those two choices.)
In the literature, you will see mention of "the" flattop window which implicitly implies that there is only one flattop window, but this is far from the truth. In fact, there are an infinite variety of flattop windows.
Now here is the proof.
We again compute the PSD of the signal containing the 11 sine waves, but now using these two Flattop windows. The lower plots show the same expanded view as before, and there is no scalloping loss visible. If we expanded the y-axis enough in the vicinity of the peak amplitude we would eventually see some scalloping loss, however, before we got to that point we would probably see bigger errors in the system (such as the ripples in the front end anti-aliasing filter for example).
Suppose in addition to the 11 sine waves that make up our signal we add 2 more low amplitude signals, the first, a 980 Hz sinusoid at -50dB and the second, a 1900 Hz sinusoid at -65dB. (These plots do not depict these extra imagined signals.) With the Flat201 we would be able to see the 980 Hz sinusoid (between two of our peaks), but not the second one because the leakage in that range would obscure that signal. However, with the Flat401 it would be reversed. We wouldn't be able to see the 980 Hz sinusoid because the main lobe is so wide there isn't enough space between the peaks to see that signal. However, the leakage is low enough at the high end of the range that you would be able to detect the 1900 Hz signal.
This demonstrates that you may not be satisfied if only a single flattop window was available.
Frequency translation is an important data acquisition technique. It's also often called zoom. (In the graphics world, zoom refers to display expansion. But frequency translation is not a display expansion.)
To those of you familiar with radio theory, it is similar to heterodyning. The samples from the A/D converter (xn) are multiplied by a complex phasor (e-jω) resulting in these two sequences (real and imaginary parts). Each of these sequences is then fed through a decimate by D low pass filter. The zoom factor (or expansion factor) is D/2. Each of these ÷D blocks represent the whole cascaded chain of decimate by 2 and 5 blocks shown before. Since two of these blocks are required, the filtering operation for frequency translation requires twice the processing horsepower as for baseband.
Since the FFT can handle complex-valued inputs, we can then feed the result of the filtering directly to the FFT and we can compute all the usual FFT-based functions (auto & cross spectra, transfer functions, etc.)
Multiplying by the complex exponential essentially twists or frequency shifts the data so that the center frequency moves down to DC. (The twisted signal is complex-valued since it is no longer symmetric around DC.) Then the filtering increases the frequency resolution by discarding the frequencies outside the band of interest.
Because of the averaging effect of the digital filters, we can often detect signals using zoom that otherwise would be buried in the noise.
For example, the upper left plot is a baseband PSD of what looks like pure random noise.
In the lower-left plot we have zoomed around 18 kHz by a factor of 10 (the resolution is improved from 100 Hz to 10 Hz). Now we can barely see that there is a sine wave buried in the noise.
Now zooming by another factor of 10 (resolution of 1 Hz) we can more clearly see the sine wave.
In the upper left is a baseband measurement with a resolution of 50 Hz. The signal looks like it has primarily one component near 5 kHz. The shape is a little broad to be a pure sine wave, but it is difficult to tell whether it is sinusoidal or random.
An increase in resolution by a factor of 10 (lower left) shows that the peak is actually a band of noise. Also, we see side bands that were not evident from the baseband display. We can also see a small glitch near 5 kHz that indicates there might be something irregular there.
So we increase the resolution again by a factor of 5 (upper right). Now we can see that this little glitch actually was two sine waves near 5.05 kHz. If we zoom by another factor of 20 (lower right) we can now see these two tones clearly.
Note that this extra resolution comes at a price. The price is acquisition time. The acquisition time is equal to the reciprocal of the resolution. So for the baseband measurement (res = 50Hz) the acquisition time is 20mS. Thus we could do 50 averages in only 1 second. The lower right plot has δf=.05Hz, so the acquisition time is 20 seconds and our 50 averages would take almost 17 minutes. This is a fact of physics; the fastest analyzer in the world can't do it any faster.
Signal generation often gets slighted in discussions of sampled data systems. However, it is just as important and just as difficult to do well. Many of the technical challenges are similar, although some are unique (e.g. the sequence generation for random noise and chirps)
The most useful output signal is the ubiquitous sine wave, the basis function for all spectral analysis. For example, to measure the distortion of a network at a specific frequency, we need to excite the network with a sine wave that is as pure as possible. The purity of the sine wave limits the levels of distortion that can be measured. In general, the dynamic range of the signal generation system should be similar to the dynamic range of the acquisition system. The sine wave can also be stepped in frequency to make transfer function measurements using the classic swept-sine technique. The other periodic functions have more occasional uses, usually for time domain measurements.
Broadband functions, the other class of output functions, are used to excite a whole band of frequencies at once. These functions allow network measurements to be made much faster than using sine waves.
Other than the sequence generation, the components of a signal generation system are similar to a data acquisition system. The interpolating digital filters are used so that the D/A and its smoothing filter can run at one fixed sample rate.
The interpolated signal can then be frequency translated to concentrate the energy of the output in a narrow band. It is common to frequency- translate the output when the acquisition system is also frequency-translating.
The D/A converts the digital numbers into a series of analog steps, each step holding its value constant for the sample period (called a zero-order hold).
An analog smoothing filter is used to smooth out these steps. The characteristics and quality measures of this filter are identical to that of the AA filters. Often the exact same filter design can be used. The D/A zero-order hold introduces a sin(x)/x frequency error which is usually compensated for with a simple one-pole filter.
Finally, the filtered output is amplified or attenuated to the desired output amplitude and buffered to yield the desired driving characteristics.
Of the sequence generators shown in the previous diagram, the most challenging one is random noise. Can a deterministic device such as a DSP chip generate a purely random number sequence? (i.e. a sequence that does not repeat) ?
No it can't. However, we can generate sequences that look random at least over a short time period. These are called pseudo-random sequences. There are many algorithms for computing pseudo-random sequences, both simple and complex. Complexity does not ensure a long sequence, however. The literature is full of examples of extraordinarily complex algorithms that yield extremely poor results.
My favorite because of its simplicity and well-researched properties in known as the linear congruential sequence. We start with an arbitrary number (R) known as a random seed. We multiply by constant a, add another constant c, and then divide by a third constant m and save the remainder from this division (R') as the next number in our random sequence. Since the remainder will always be between 0 and m, the random numbers generated are also in this range. If the sequence covers all such values, then it is called a maximal length sequence. Choosing the three constants according to these two simple constraints has been proven to be sufficient to ensure a maximal length sequence. Thus we can make our sequence as long as desired by picking a large value for m. Choosing m to be a power of 2 simplifies the arithmetic. m = 216 is good enough for some applications. 232 is much better, but requires a 32 by 32-bit multiply which may require multiple precision arithmetic in some implementations.
The name Dynamic Signal Analyzers was coined by HP, one of the early makers of such equipment. Other manufacturers have sometimes used this fairly descriptive name, although FFT analyzers or Fourier analyzers is also common. They are designed to analyze signals and systems covering a broad range of fields and applications, usually involving something that moves (mechanical) with some applications involving purely electrical systems (e.g. telecommunications).
The measurements are usually divided into these three categories.
For time domain measurements, the sophistication of a dynamic signal analyzer is not usually required. A digital scope would suffice. However, these functions are useful, since it often means that you can leave the scope behind.
Most spectral analysis problems can be accomplished with a single input channel, and draw on the power of the FFT to display the time histories in the frequency domain.
Network measurements refer to measuring the input/output characteristics of a system usually to compute a frequency response function (FRF) or a transfer function (mathematical system model). Network measurements usually require a signal generator to excite the system under test, and always require two input channels, one to measure the excitation and one to measure the system response.
Dynamic signal analyzers cover such a diverse range of applications that it is somewhat difficult to categorize them. However at least 90% of the time I find that the customer's application can fit into one of the following categories.