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Circuit Surgery Regular Regular clinic clinic by by Ian Ian Bell Bell Topics in digital signal processing – an introduction to processing digital signals W e are looking at various topics related to digital signal processing (DSP). DSP covers a wide range of electronics applications where signals are manipulated, analysed, generated, stored or displayed as digital data but originate from and/or are converted to real-world signals for interaction with humans or other parts of the physical world. Fig.1 shows the key elements of a generic DSP system with a signal path from an analog input via digital processing to an analog output. This does not necessarily represent every DSP system (not all have all the parts shown), but it serves are as reference for the various subsystems we will consider. In previous articles, we have concentrated on the analog parts of the system (filters and sampling), and on the data converters. This month we will take a look at some basics of digital/sampled signal processing and look at a simple example: a moving average filter. Sampled signal processing Consider sampling a continuous signal xc that is function of time (t). We can write is as xc(t). Assume that the sampling period is T. The sampled signal xs is also a function of time, so we could write xs(t), but the samples only exist at certain times, specifically integer multiples of T. That is, t=nT, where n=1,2,3 etc. So we often write a sampled signal as xs(nT). Sometimes, this written as xs(nΔT). The Greek letter delta (Δ) emphasises that T represents a time period rather than a continuous time value. Using nT defines a specific sample time, but n alone is sufficient to identify the sample, so we often just use the sample index and write sampled signal as x(n). We can drop the Analog In Antialiasing filter Sample and hold s subscript as the sample-number indexing implies a sampled signal. In general, we assume that n ranges from −∞ to +∞ (∞ = infinity). This defines a signal which lasts for all time, which is common in theoretical analysis. Of course, real signals have finite duration. In this discussion, we will only consider systems operating with one sample period, unlike the oversampling DACs discussed last month. In signal processing, we typically need to consider specific points in time relative to the current time and often define n (or some other symbol) to represent the current sample point of the signals. So, for example, the input to a system at the current time would be x(n) and the current output y(n), as shown in Fig.2 (the Digital Processing part in Fig.1). The previous input sample is x(n-1), the input two sample periods before the current time is x(n-2) and so on. Similarly, the next value of the input, one sampling interval into the future, is x(n+1), followed x(n+2) and so on. This is illustrated in Fig.3, which shows the samples around the current time in what we assume is a long-duration signal. The only sample value directly available to the processing system is the current input value, x(n). However, we cannot do much that’s useful with just the current sample. We also need access to earlier samples to implement useful functions such as filters. To make use of earlier sample data, it must be retained in a memory or digital registers in a circuit. The sample number (n, n-1, n-2) then indicates the memory location or specific register holding a particular sample relative to the current input value. Storing sample data in memory is equivalent to delaying the signal by multiples Digital ADC Digital processing Analog DAC Fig.1: a generic digital signal processing (DSP) system structure. Practical Electronics | December | 2024 Reconstruction filter of the sampling period. Fig.4 shows the signal in Fig.3 delayed by one sample period from the perspective of the same current time. The value of the delayed signal at the current time is x(n-1). If we delayed the signal by two sample periods, sample x(n-2) would be available at the current time. In general, x(n-m) represents the signal x delayed by m sampling periods. A signal processing system may also need to use previous output values in the calculations it performs. Memory or registers can be used to store these values in the same way as input values. Fig.5 shows a generalised digital signal processing system; the input and output are the signals x and y, respectively, at current sample point Digital y(n) Fig.2: a digital sample processing scheme. x x(n–2) x(n–1) x(n) x(n–3) x(n+1) T Current time t x(n+2) Fig.3: part of a sampled waveform. x x(n–2) x(n–4) x(n–3) x(n–1) x(n) x(n+1) t Current time Fig.4: like Fig.3 but delayed by one sample. x(n) y(n) x(n–1) x(n–2) Memory Out Digital processing x(n) x(n–3) x(n–m) y(n–1) Processing y(n–2) y(n–3) Memory y(n–p) Fig.5: a general digital processing system. 49 specific sample number a result is associated with is important (ie, it avoids time-shifting large changes). An LTspice implementation y ( n )= x ( n−4 ) + x ( n−3 ) + x ( n−2 ) + x ( n−1 ) + x ( n ) 5 | | ( ff ) N sin π ( ff ) sin Nπ s s Fig.6: this shows how I implemented signal delays in LTspice. n [that is, x(n) and y(n)]. A total of m previous inputs [x(n-1) to x(n-m)] and p previous outputs [y(n-1) to y(n-p)] are stored in memory and used by the signal processing. Real systems vary widely in the number of previous input values used, and many do not use any previous output values. It is not uncommon for the fundamental theory related to signal processing to involve future input values, denoted [x(n+1), x(n+2), ...]. Systems defined in terms of future inputs are called noncausal systems; obviously, they cannot be implemented for real-time input signals. However, theoretically developed noncausal systems can be modified to make them causal; for example, by time-shifting the calculation so that only current and past values need to be used. If the signal being processed is already stored in its entirety (eg, digital audio on a hard drive), then with respect to any given data point, the ‘future’ values are available. A moving average filter Possibly the simplest digital signal processing function is the moving average filter – we will use it as an example for that reason. It is worth quickly investigating its properties first. Moving averages are widely used in a variety of fields, not just electronics. If you do an internet search on “moving average”, the top results are likely to mainly relate to finance and economics. Moving averages are useful because they provide a simple and clearly defined smoothing function for data. Plotting a moving average makes it easier to visualise trends. Moving averages can also help to remove known periodic fluctuations. An example of this is the use of seven-day moving averages for the number of cases during the COVID pandemic. Test labs may not be operating every day, which creates peaks and troughs over the course of a week. The seven-day moving average removes this effect, as well as smoothing out the natural random variations in case numbers. In electronics, moving average filters 50 are very effective at removing random noise from signals. They are particularly useful when the signals have fast step-changes in level (eg, pulse-based signals), where the timing and shape of steps needs to be preserved as much as possible through the filtering process. They are not very useful for frequency separation (rejecting some frequency ranges while passing others). As its name suggests, a moving average filter finds the average (mean) of a range of sample values from the input to obtain the output. This is achieved by summing the samples and dividing by the number of samples. For example, for five samples, the moving average processing function is: y ( n )= x ( n−4 ) + x ( n−3 ) + x ( n−2 ) + x ( n−1 ) + x ( n ) 5 | | ( ) ( ) f The equation sin relates Nπ to Fig.5, in which m=4 and in which nof sprevious output values (y) are stored fand used – only N sin π fs the input memory is present. This is a causal version that can be implemented in real time. We could also take the average of the current sample, two past and two future ones. This average is centred on the current sample, but is non-causal. However, this may be useful in data analysis if the In Circuit Surgery, we often simulate circuits using LTspice, so it is worth checking if we could do this for a moving average filter. It would be useful to see the effect on a noisy waveform, and to obtain frequency response curves. This is perhaps not an obvious thing to use LTspice for, but it is worth trying, and doing so also demonstrates some useful features of the simulator. We can use a behavioural source to implement the averaging maths if we can produce signals to represent the stored previous waveform values. As discussed above, storing previous samples is equivalent to delaying the sampled signal by an integer multiple of the sampling period. LTspice behavioural sources support a delay function: delay(x,t), which delays signal x by time t, enabling us to implement the memory function. An example of this is shown in Fig.6. In the five-sample moving average equation above, if n=4, the terms in the equation are x(0), x(1), x(2), x(3) and x(4), where x(0) is the oldest sample (after 4 sample period delays); the current input signal is x(4). These index values are used to name the signals in the LTspice schematic in Fig.6 (nets x0 to x4). Behavioural sources B1 to B4 implement the delays. Each delay is the same; this value is defined as a parameter, sp (for sampling period) using the .param directive. This means we only have to change this one value to change the delays of all the behavioural sources. LTspice parameters are used by putting the parameter name (or an expression using them) inside curly braces (eg, {sp}). This is used in the equations for the behavioural sources, for example, Fig.7: the results of running the simulation on the schematic shown in Fig.6. Practical Electronics | December | 2024 B4 generates the voltage on net x3 as a copy of the voltage on net x4, delayed by sp using V=delay(V(x4),{sp}). In the Fig.6 circuit, behavioural source B5 creates the output signal y by implementing the five-sample averaging function: V=0.2*(V(x4)+v(x3)+V(x2)+V(x1)+ V(x0)). The input (x4) is a sinewave generated by voltage source V1. The results are shown in Fig.7, which shows the evenlyspaced delays on the waveforms. The output settles to a sinewave because the averaging function is performing a linear operation on its sinewave inputs. However, the waveform is initially not sinusoidal because, at the start, the averaging uses the initial values of the delayed waveforms, which are zero rather than sinusoidal. It is common in digital signal processing for the outputs to not be meaningful until all the sample memories have values relating to the actual input signal in them. Usually, this is not a problem, and designs are based on continuous processing of a long-lasting signals. The circuit in Fig.6 is missing the filtering and sampling parts of a typical DSP system, as per Fig.1. These can be added using the “sample” device we discussed previously to produce a sampled waveform, along with behavioural filter devices to provide the anti-aliasing and reconstruction filters. These were discussed in previous articles in this DSP series. Fig.8 shows the components added to the schematic in Fig.6 to implement sampling and filtering. Sinewave source V1 is now connected to the net in, with net x4 (current sample) now coming from the output of the sample-and-hold block. All the behavioural sources from Fig.6 (B1 to B5) are also present in the simulation schematic, but are not shown in Fig.8. In the Fig.8 circuit, the sampler (A1) and filter devices (U1 to U4) use the sampling period parameter value to control their operation. The sampling clock period produced by source V2 is equal to sp and the cutoff frequency of the filters is set to the Nyquist frequency (half the sampling frequency) at 1/(2*sp). The results from simulating Fig.8 are shown in Fig.9. The upper plot pane shows the sampled sinewaves with different delays. If you look at the green waveform (x4, the current input to the processing) at a given sample point, you can see the same sampled level appearing on the other waveforms in sequence (red, cyan, magenta, yellow) as the value moves through the delay chain. The lower pane shows the sampled output (y) and the final output after passing through the reconstruction filter (y_filt). The filtered output is very similar to the output of the non-sampled version of the system in Fig.7. Practical Electronics | December | 2024 Fig.8: new and modified components that add filtering and sampling to the circuit in figure 6. B1 to B5 are also present but not shown here Fig.9: the results from the schematic shown in FIg.8. Fig.10: the AC analysis results from the schematic shown in Fig.8. AC analysis failures Given that the moving average function acts as a filter, we might want to look at its frequency response. Typically, we would use AC analysis in LTspice to achieve this. The V1 source in Figs.6 & 8 is configured ready for this, with an AC amplitude of 1V (AC 1). All we need to do is configure the AC analysis, for example, using .ac dec 50 100 5k to plot the response from 100Hz to the Nyquist frequency of 5kHz. The sampling period (sp) is 100μs, so the sampling frequency is 1 ÷ 100μs = 10kHz. The results are shown in Fig.10. The upper trace is for in_filt and therefore shows the response of the anti-aliasing filter. This is as expected from the setup of the behavioural filters, and shows very little attention at the 1kHz frequency of the sinewave from V1 used as the input of the simulation in Figs.7 & 8. The lower trace, the output of the signal processing (y), is zero at all frequencies. The AC analysis is not working because it is not supported by the sample-and-hold device. The switched nature of the sampling means that we cannot use basic AC analysis to determine the frequency response. 51 Fig.11: the FRA device and probe as added to the Fig.8 circuit. Frequency Response Analysis LTspice provides an alternative to AC analysis called Frequency Response Analysis (FRA). AC analysis uses a linearised model of the circuit to find the frequency response, which is fine for analog filters and amplifiers, but does not work for switched circuits. FRA was added to LTspice primarily to assist with the stability analysis of switch-mode power supplies (SMPS). It can help to find the loop gain of a feedback system but it has other uses. It works by automatically running multiple transient simulations at different frequencies, measuring waveform amplitudes and calculating gains.This means the fully operational circuit is analysed, not just a linearised version, so any switching (or in this case sampling) operations operate normally. We discussed LTspice’s FRA in the March and April 2024 Circuit Surgery Fig.13: initial results from the FRA. columns, where we illustrated the basic principles by analysing an op amp circuit feedback loop. The FRA measurement is controlled by a special FRA device, which is usually inserted into the feedback loop of the circuit. This device then injects signals and measures the response of the loop. There is no feedback loop in the Fig.8 circuit, so we cannot use the FRA device on its own in this way. We can insert it at the input to inject the test signals, but as it is not in a loop, the response cannot be measured from there. Fortunately, LTspice provides another special device called an FRA probe that works in conjunction with the FRA device to make the measurements. It is aimed at complex SMPS circuits that the FRA cannot handle on its own. However, it works for a simple input-to-output frequency response analysis. The FRA probe has two differential inputs – one for the input and one for the output signal, with both used for the response analysis. To perform the analysis, we add two components: the FRA device and the FRA probe device to the schematic in Fig.8 (remember the whole circuit also includes B1 to B5 from Fig.6). This is shown in Fig.11 where the component named <at>1 is the FRA device Fig.12: setting up the FRA stimulus. injecting the test signals and the component named &1 is the probe making the measurements. The input sensing part of the probe is connected to the output of the FRA device on net in_FRA. The output sensing part of the probe is connected to the output of the reconstruction filter on net y_filt. The net at the input to the U3 filter is also changed to be in_FRA rather than in (not shown in Fig.11) to connect the FRA device’s test signal to the system input. This leaves the in net from V1 not connected to anything, but this is not a problem. To run the FRA, we need to set the analysis configuration to Transient Frequency response and place the .fra directive on the schematic. The way that the FRA operates is configured by rightclicking the FRA device. Fig.12 shows part of the dialog which appears after doing that. Here, we have set the start and end frequencies for the analysis and selected four points per octave. This controls which frequencies of sinewave will be used in the simulation. A low start frequency is used to facilitate linear frequency plotting but the “Coarse Steps” setting ensures that not too much time is used simulating low frequency signals. The analysis time settings ensure that enough time is given for the circuit to settle after frequency changes. If these settings are not used, the results are poor at the high-frequency end Simulation files Most months, LTSpice is used to support descriptions and analysis in Circuit Surgery. The examples and files are available for download from the PE website: https://bit.ly/pe-downloads Fig.14: FRA results using more data points. 52 Practical Electronics | December | 2024 Fig.15: the theoretical response of a five-point moving average filter with a 10kHz sampling frequency. of the range, but the values here are not necessarily optimal. Everything else is left blank (default). For more discussion on the FRA settings, see Circuit Surgery, April 2024. There is a checkbox at the bottom of this dialog to disable the FRA device. That’s important as the FRA device must be enabled to run the FRA and disabled when running other types of analysis. FRA results The initial results from the FRA (for output y_filt) are shown in Fig.13. The text about phase margin is automatically added but is not relevant here – it is easily deleted. Fig.16: the results from the FRA plotted on linear axes so that we can compare them directly to ) Fig.15. + x ( n−3 ) + x ( n−2 ) + x ( n−1 ) + x ( n ) x ( n−4 y ( n )= The results are much better than in Fig.10, showing the FRA is working, but the curve lacks detail. This is because the FRA has a maximum of four data points per octave (as per Fig.12), which is insufficient for this graph. This is a limitation of the FRA in this context, but it can be overcome by adding frequency points in the “Add These Specific Frequencies[Hz]:” box. The improved result in Fig.14 was obtained by adding about 25 extra data points in the 1.5kHz to 4.9kHz range. The theoretical frequency response (the gain magnitude at frequency f) from a N-sample moving average filter is given by: 5 | | ( ff ) N sin π ( ff ) sin Nπ s s As usual, fS is the sampling rate. The gain is 1 at f=0. This is plotted use Excel in Fig.15, and the FRA results are plotted on linear axes in Fig.16 (the same data as in Fig.14). The results are similar, but the FRA shows lower gain than the pure moving average function at higher frequencies. That is due to the action of the antialiasing and reconstruction filters. We will continue to look at signal processing next month. 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