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Sampling and Time Division Multiplex Objectives To understand the concept of sampling a continuous analogue waveform To investigate sampling a waveform using an analogue to digital converter To investigate the effects of sampling rate and to understand the concept of aliasing To appreciate the Nyquist limit applied to sampling rate To investigate time division multiplexing of signals Practical 1: Sampling Analogue Signals Objectives and Background Sampling In this practical you will investigate the effects of sampling a continuous analogue signal using an analogue to digital converter. Read the Resources section on Sampling if you do not understand the process of sampling an analogue signal, or parameters such as resolution and sampling rate. The set-up for the practical is a sinusoidal analogue signal digitised by an analogue to digital converter (A/D) at a constant rate of 20 kHz and then passed back out to a digital to analogue converter (D/A). The frequency of the sinusoidal signal can be varied, so the effect of the ratio between signal and sampling rate can be observed. The resolution of the A/D and D/A is 8 bits (i.e. 256 levels). In the practical you will change the resolution to 4 bits (16 levels) and 2 bits (4 levels) to see the effect. You will also see from the resulting waveshape that, at first glance, it is difficult to tell whether a signal is being sampled at insufficient resolution or insufficient sample rate. The A/D and D/A are part of the on-board microprocessor system on the hardware. The data is passed through the microprocessor, where the resolution is changed as required. The functions of the on-board microprocessor are controlled by commands automatically sent by you when you start the practical or press a button on the practical diagram. Practical 1: Sampling Analogue Signals Perform Practical Use the Make Connections diagram to show the required connections on the hardware. Identify the Micro Controller and A/D – D/A circuit block, located towards the top, lefthand corner of the board. Associated with this circuit block, set the A/D 1 Offset, the A/D 1 Amplitude and the D/A 1 Offset to mid position. Set the Function Generator to Slow. Set the Signal Level Control for maximum output. Open the frequency counter and set the Frequency (in the Function Generator block) to approximately 400 Hz. Close the frequency counter and open the oscilloscope. In the Function Generator block, use the waveform selector to select a sine wave output. On the oscilloscope, note that the output signal is very similar to the input signal and that the system is set to 8 bit resolution Increase the size of the oscilloscope so you can see the waveforms more easily. Change the resolution to 4 bit and notice that the output has more steps in it. Now try 2 bit resolution and note that the output contains only a few discrete levels. Try the different resolutions and also adjust the amplitude of the signal using the Signal Level Control. Note that at 2 bit resolution most of the signal waveshape is lost. With 2 bit resolution, change from sine to triangle waveform and note that it hard tell the difference. Return to a sine wave and select 8 bit resolution and maximum amplitude. Now increase the frequency of the function generator gradually. You will need to increase the timebase speed on the oscilloscope so you can see only a cycle or two to see what is happening. Note that the waveshape has steps in it now. This is because the signal frequency is such that there is only time to take a few samples in each cycle. Note that the effect on the output is similar to reducing the resolution. If you increase the frequency too far, some strange effects occur as a result of aliasing. This is examined further in Practical 2. Practical 2: Aliasing Objectives and Background The Effects of Aliasing In this practical you will investigate the effect of sampling an analogue signal at sample rates near to and below its frequency. Aliasing can be a significant problem in any sampling system and can result in completely misleading results. The lowest rate that can be used to sample a signal is twice the frequency of the signal you are trying to sample. Even then the results may not be satisfactory. For example, if you sampled a sinusoidal signal at twice its frequency and looked at the result all you would see is that the signal is one level during one sample and another level during the next sample. This may be all you need to know, as it does convey the frequency of the signal - but all the other detail of the signal has been lost. A sampling rate at twice the signal frequency is called the Nyquist limit. You may wonder what happens beyond this limit (sampling at less than twice the signal frequency) and you might be thinking that you get nothing out. This would be rather satisfactory but, in reality, you get waveforms out that imply the frequency is below the Nyquist limit. This is rather like a multiplying or mixing process using the sampling rate at the multiplying signal. This effect is called aliasing, because the waveform you get is not real and is an “alias” of the frequency being sampled. There is more detailed information on this quite complex problem in the section on aliasing. The important thing is to recognise that aliasing can happen; to recognise when it does and not to be misled by its effects. In the Practical you will be able to see aliasing at work. Interestingly, there are situations when the effect can be used to digitise a high frequency signal. This is called sub-sampling, but is outside the scope of this practical. In the practical, only a single frequency signal is used; but in reality the signal being sampled may contain many frequencies. The Nyquist limit says that you must sample at twice the highest frequency present in the signal. In some cases some of the higher frequencies may not be of interest but, to prevent them appearing as aliases, a low pass filter with a cut-off at half the sampling frequency is used. This is sometimes referred to as an “anti-aliasing filter”. Practical 2: Aliasing Perform Practical Use the Make Connections diagram to show the required connections on the hardware. The hardware setup used is similar to that used in Practical 1. In this Practical you will only be using 8 bit resolution. Set the A/D 1 Amplitude, the A/D 1 Offset and the D/A 1 Offset to mid position. Set the Function Generator to Fast. Set the Signal Level Control for maximum output. Open the frequency counter and set the Frequency to approximately 2 kHz. Open the oscilloscope. In the Function Generator block, select a sine wave. Note that the output signal has some steps due to the sampling rate (20 kHz) being only 10 times the signal frequency, which means that there are only 10 samples per signal frequency cycle. Increase the signal frequency and note that the sampled signal becomes more and more ragged. Near to the Nyquist limit (10 kHz) notice that rather strange things start to happen. It is possible to sample at the Nyquist limit but here the results are difficult to interpret. This is because the sampling rate and signal are not synchronised. Note that, very near to 10 kHz, the amplitude of the waveform appears to vary at a lower frequency. As you will see from a later Assignment, this waveform resembles a double sideband suppressed carrier signal. This confirms that sampling is a multiplying process. Set the frequency to about 9.5 kHz. Move the frequency counter probe to the output of the D/A Converter (monitor point 2). Now, slowly raise the frequency. As the frequency is raised above 10 kHz note that frequencies appear below 10 kHz. These are aliases. Set the signal frequency to 15 kHz (you will need to move the frequency counter back to monitor point 1, temporarily). Note the result on the oscilloscope. Move the counter back to the sampled output signal and measure the frequency. How do you think it is related to the input frequency? Notice, also, that further effects occur above the sampling frequency (20 kHz). Practical 3: Time Division Multiplex Objectives and Background In this practical you will investigate time division multiplex using two A/D converters and a single D/A converter. Two analogue signals: one a sinusoid and the other a variable dc voltage are fed into the two a/d converters. The microprocessor samples the two alternatively at 20 kHz. The multiplexed signal is passed to a D/A and you can see it on the oscilloscope. Note that, if the overall sample-rate is 20 kHz, then for two signals the sampling rate is 10 kHz, with the associated problems of this lower sampling rate. Practical 3: Time Division Multiplex Perform Practical Use the Make Connections diagram to make the required connections on the hardware. Set the A/D 1 Amplitude and A/D 2 Amplitude to maximum. Set the A/D 1 Offset, the A/D 2 Offset and the D/A 1 Offset to mid position. Set the Function Generator to Slow. Set the Signal Level Control for maximum output. Open the frequency Counter and set the Function Generator Frequency to 1 kHz. Open the voltmeter and use it to set the variable dc Source to approximately zero. Open the oscilloscope. On the Function Generator block, select a sine wave output. Note the signal on the upper trace, containing samples of the sine wave alternating above and below the dc voltage. Adjust the dc Source voltage and note that the upper trace changes position relative to the zero volt line, but its waveshape remains constant. Adjust the Function Generator frequency of the sine wave and confirm that the Nyquist limit is about 5 kHz.