Synchronous Time and Frequency Domain Analysis of Embedded Systems

Editor's Notes

• #6: Any real waveform such as that measured on an oscilloscope can be separated into the sum of orthogonal functions. The most common set of such functions are sine and cosine waves. Here we show a waveform on the left side of this slide that is composed of the sum of 2 sine waves of different frequencies as an example. The Fourier transform breaks waveforms down into their sine wave components such that it represents the original waveform as a set of complex numbers representing the magnitude and phase of each one of the constituent sine waves. The waveform is known as the frequency spectrum or simply the spectrum.
• #7: One common way to measure the frequency spectrum of a signal is to use a spectrum analyzer. This instrument, a very simplified block diagram shown here, determines the magnitude of each frequency component by sweeping a measurement filter over a range of frequencies and measuring the power at each frequency. This instrument has the benefit of being very accurate due to its low noise and can measure over a very wide frequency range. The measurement is not real time in the sense that the instrument measures frequency over the sweep time. The sweep time itself is dependent on many factors including the frequency span. In general, however, the sweep speed will be fast over smaller spans. Many spectrum analyzers also include a Fourier transform that allows real time measurement in spans of 40 MHz.
• #11: Time domain samples are transformed to N FFT /2 equally spaced lines in the frequency domain. The number of time domain samples times the sample interval of the A/D converter determines the integration time over which each frequency component is measured. The frequency resolution is dependent on this integration time such that the resolution is the inverse of the integration time or, equivalently, the A/D sampling rate divided by the number of samples over which the FFT is computed. The total frequency span of an FFT is determined by the sampling rate of the FFT and covers a span from DC to Ts/2 when the sampled waveform is real. Each frequency sample computed by the FFT can be viewed as a “bin” whose shape in the frequency domain is determined by the integration window over which the bin value is computed. Because all of the samples in the integration are weighted equally, this shape is that of a sin(x)/x function. The bins overlap in the frequency domain in such a way that the amplitude of each bin contains some energy from the adjacent bins.
• #12: There are two important rules when working with FFT’s: The resolution bandwidth is determined by the integration time so, for example, a 1 Hz resolution requires an integration time of 1 second and a resolution of 10 Hz requires 100 ms. The highest measurable frequency is set by the A/D sampling rate with the maximum frequency equal to ½ of the sampling rate. Meeting these requirements often requires very long time domain records if, for example, one wants to measure a high resolution over a relatively narrow span at a high center frequency. For example, measuring a 100 MHz span centered at 1.5 GHz (from 1.45 GHz to 1.55 GHz) with a 100 Hz resolution will require 50 million A/D samples at 5 Gs/s since we must sample fast enough to acquire the 1.5 GHz center frequency and long enough to resolve the 100 Hz resolution.
• #13: Another limitation of the FFT in spectrum analysis applications is the inefficiency of the computation for analysis of narrow spans at high frequencies. A conventional FFT is computed on a single record of data samples acquired by an A/D converter and the spectrum is computed from DC to ½ the sample rate. The desired resolution bandwidth is set by the number of time samples used in the computation and the span is set by zooming in on the desired section of the spectrum. A better approach is to select the desired center frequency using a down converter to shift the center frequency of the span of interest down to DC and then sampling this bandwidth limited spectrum at a lower rate consistent with the span of interest. The lower sampling rate allows a much narrower resolution bandwidth using a smaller number of time domain samples. The shorter time record improves the computational throughput dramatically.
• #14: This slide shows a more detailed comparison of the two methods of computation. The conventional FFT is computed on the complete data record and then zooming in on the desired span. The R&S RTO uses a digital down converter to shift the center frequency and span of interest to DC and reduce the sample rate at its output. The window function and FFT is computed on the shorter time record at a much faster rate. Zooming is also possible after the FFT.
• #15: The fast computation of the FFT that results from the use of the down converter can be exploited in overlap processing. Conventional oscilloscopes compute FFT‘s on time records that are separated in time by an amount equal to the processing time for each FFT. Overlap processing allows the computation of multiple FFT‘s on overlapping data records in such a way that data making up one FFT consists of a percentage of the data from the previous FFT with the remaining part being new data. The percentage of overlap can be set by the user. In this slide, a 50% overlap is shown. Overlap processing is valuable because most signals are dynamic and change over time and, as you recall, the window function attenuates the signal at the edges of the inteval. Intermittent signals that appear near the edges of the interval will not be seen due to the weighting of the window but with a 50% overlap, the signal attenuated in one FFT will be seen in the next one.
• #16: A non-integral number of cycles cause discontinuities in the waveform produce high frequency transients which add false frequency information => Leakage Multiply the waveform record by applying a window function that is zero at the both ends, reducing the discontinuities, resulting a more accurate FFT measurement.
• #17: A non-integral number of cycles cause discontinuities in the waveform produce high frequency transients which add false frequency information => Leakage Multiply the waveform record by applying a window function that is zero at the both ends, reducing the discontinuities, resulting a more accurate FFT measurement.
• #28: In the frequency domain, however, the affect is much more obvious. While the interleaving artifacts appear as noise in the time domain they are, in fact, not random and appear as discrete spectral lines. These frequency components appear at the sampling frequency of each A/D and at harmonics of this frequency. For example, if a 10 Gs/s A/D is built from four 2.5 Gs/s interleaved A/D converters, there will be spectral components at 2.5, 5, 7.5, and 10 GHz. These spectral components then mix with the spectrum of the input signal to generate additional spurs. The level of these spurious frequency components is fairly low – typically 50 dB below full scale. Since it is impossible to separate the spurious from the signal spectrum, the noise floor is essentially at the spurious level. This can be seen in this slide where on the left we see the spectrum of a 100 MHz sine wave sampled with interleaving and without interleaving on the right. The fundamental and harmonics of the sine wave are clearly seen in both spectra but the spurious in the spectrum on the left limit the effective dynamic range to 50 dB while the non-interleaved A/D produces a spectrum with 74 dB dynamic range.
• #29: Adding an additional low noise amplifier to the front end of the instrument ahead of the A/D improves the noise from -143 dBm/Hz to -148 dBm/Hz. This additional gain enables measurement of very small signals. Also note the very low spurious even with the lower noise floor.