YEDITEPE UNIVERSITY ENGINEERING FACULTY COMMUNICATION SYSTEMS LABORATORY EE 354 COMMUNICATION SYSTEMS EXPERIMENT 3: SAMPLING & TIME DIVISION MULTIPLEX (TDM) Objective: Experimental verification of the sampling theorem; sampling and message reconstruction. Time division multiplex (TDM) and its recovery. Equipment: Twin Pulse Generator Module Audio Oscillator Module Master Signals Module Dual Analog Switch Module Multiplier Module Headphone Amplifier Module Integrate & Dump Module Oscilloscope General Information: Sampling is the first step of the transformation of an analog signal into a digital format. Before it is possible to transmit analog information via a digital system the analog signal must be first transformed into a digital format. Sampling can be made by two types of procedures. In natural sampling, a slice of the waveform is taken and thus, the shape of the top of each sample is the same as that of the message. In sampling with sample and hold, called also as flat top sampling, a slide of the waveform is taken but the top of the slice does not preserve the shape of the waveform. These two sampling types are compared in Figure 4.1. Figure 4. 1 Natural Sampling (above) and Flat Top Sampling (below) 1/12
The arrangement to take samples of a sinewave is shown in Figure 4.2 Figure 4. 2 Sampling a Sine Wave The natural sampling is shown in Figure 4.3 and the sampling by sample-and-hold is shown in Figure 4.4. Figure 4. 1 Natural Sampling Figure 4. 4 Sampling by Sample-and Hold 2/12
Mathematical Review of Sampling Using elementary trigonometry it is possible to derive an expression for the spectrum of the sampled signal. The sampled signal is; sampled signal y(t) = m(t). s(t) (4.1) where, m(t) = V. cosμt is the message signal as a single cosine wave applied to the input of a switch having the switching function s(t) which is shown in Figure 4.5 and is expressed analytically by the Fourier series expansion as; s(t) = a 0 + a 1 cos 1 ωt + a 2 cos 2 ωt + a 3 cos 3 ωt + (4.2) s(t) is a periodic function of period T = (2.π)/ω.sec. When s(t) has the value 1 the switch is closed, and when 0 the switch is open. Figure 4. 2 The Switching Function Expansion of s(t), using equations (4.1) and (4.2), shows it to be a series of DSBSC signals located on harmonics of the switching frequency ω, including the e zeroeth harmonic, which is at DC, or baseband. The magnitude of each of the coefficients ai will determine the amplitude of each DSBSC term. The frequency spectrum of this signal is illustrated in graphical form in Figure 4.6. Figure 4. 3 The Sampled Signal in the Frequency Domain Inspection of Figure 4.6 reveals that, provided ω 2µ, there will be no overlapping of the DSBSC, and, specifically, the message can be separated from the remaining spectral components by a low pass filter. This means that, if a signal is band limited, i.e., if its Fourier transform is zero outside a finite band of frequencies, and if the samples are taken sufficiently close together in relation to the highest frequency present in the signal, then the samples uniquely specify the signal, and we can reconstruct it perfectly. This result is known as the sampling theorem. This theorem also says that the slowest usable sampling rate is twice the highest message frequency (ω 2µ). 3/12
Reconstruction / Interpolation From Fourier series analysis, and the consideration of the nature of the sampled signal, it can be seen that the spectrum of the sampled signal will contain components at and around harmonics of the switching signal and the message itself. Thus, a low pass filter can be used to extract the message from the s samples. The reconstruction circuitry is illustrated in Figure 4.7. Figure 4. 4 Reconstruction Circuit If the reconstruction filter does not remove all of the unwanted components-specifically the lower sideband of the nearest DSBSC, then these will be added to the message. Thus, the unwanted DSBSC was derived from the original message. It will be a frequency inverted version of the message, shifted from its original position in the spectrum. The distortion introduced by these components, if present in the reconstructed message, is known as aliasing distortion. If message reconstructions by low pass filtering of natural samples results in no distortion, then there must be e distortion when flat top pulses are involved. The pulse width determines the amount of energy in each pulse, and so can determine the amplitude of the reconstructed message. But, in n a linear and noise free system, the width of the samples plays no part in determining the amount of distortion of a reconstructed message. Time Division Multiplex (TDM) In this experiment we will sample several messages, and their samples will be interlaced to form a composite, or Time Division Multiplexed TDM signal. In Figure 4.8 the composition of a 2-channel TDM signal is shown. If two messages were sampled, at the same rate but at slightly different times, then the two trains of samples could be added without mutual interaction. 4/12
Figure 4. 8 Composition of a 2-channel TDM At the transmitter, there are some conditions for producing TDM signal. Considering 2- channel TDM signal composition shown in Figure 4.8; the width of the samples is δt, and the time between samples is T. The sampling thus occurs at the rate (1/T) Hz. One sample from each channel is contained in a frame, and this is of length T seconds. In principle, for a given frame width T, any number of channels could be interleaved into a frame, provided the sample width δt was small enough. At the receiver, an arrangement to see how the samples from one or the other channel could be separated from the TDM signal is called a de-multiplexer. An example is illustrated in Figure 4.9, for doing this, it is provided that the timing information was available a knowledge of the frame period T and the sampling width δt. Figure 4. 9 TDM Demultiplexer Block Diagram 5/12
The switching function s(t) has a period T. It is aligned under the samples from the desire channel. The switch is closed during the time the samples from the desired channel are at its input. Consequently, at the switch output appear only the samples of the desired channel. From these the message can be reconstructed. To recover individual channels it is necessary to have a copy of the sampling clock. This is generally derived from the TDM signal itself. The TDM signal contains no explicit information to indicate the start of a frame. Channel identification is of course vital in a commercial system, but we can dispense it for this experiment. A PCM encoder is shown in Figure 4.10. Here, samples are coded into binary digital words, and placed into frames of eight slots, each slot being of length equal to a bit clock period. Each frame contained a coded version of a flat top sample of an analog signal (obtained with a sample-and-hold operation), together with a frame synchronization bit. Figure 4. 10 PCM Encoder Timing Diagram If the contents of every alternate frame were removed from the serial data, then it would appear that the sampling rate had been halved. Then, the allowable bandwidth of the signal to be sampled would have been halved. The message could still be decoded if each alternate frame could be identified. Thus, the empty spaces in the data stream could be filled with frames derived by sampling another message. These would not interfere with frames of the first message. Thus, two messages could be contained in the one data stream. This is a Time Division Multiplexed Pulse Code Modulated (PCM TDM) signal. 6/12
Procedure: 1. Build the sampling system shown in Figure 4.2 for natural sampling. In order to generate the switching function in the figure use the TWIN PULSE GENERATOR module. Take the sampling frequency as 8.333 khz. Observe the message signal and its sampled version on the oscilloscope and draw them on a scope sheet. 2. Recover the message signal by using the LPF output of the HEADPONE AMPLIFIER module which has the response of a 3 khz LPF. Draw the recovered signal on a scope sheet. 3. Change the frequency of the message signal within the full frequency range of the module used for message signal generation. Explain the significant changes, if any, on the reconstructed signal as the message frequency changes. 4. Observe the changes when the width of the switching pulses is changed. Comment on the results. 5. To realize flat top sampling, use the INTEGRATE & DUMP module. Draw the waveforms of the message signal and the sampled signals on a scope sheet. 6. Use the TUNABLE LPF module to recover the message signal. Turn the tune knob of the TUNABLE LPF within its full range and explain the significant changes, if any, on the reconstructed signal. 7. Use 2 khz sinusoidal signals to observe a 2-channel TDM signal on the oscilloscope and draw the waveform on a scope sheet. Take the sampling frequency as 8.333 khz. 8. Change the time and the width of the samples of the input signals and draw again the waveforms on a scope sheet. 9. Build the circuit of Figure 4.9. Observe the resulting waveform on the oscilloscope and draw it on a scope sheet. 10. Pass the TDM signal through a LPF and observe the result on the oscilloscope. Draw the scheme on a scope sheet. 7/12
YEDITEPE UNIVERSITY DEPARTMENT of ELECTRICAL & ELECTRONICS ENGINEERING E X P E R I M E N T R E S U L T S H E E T Course: EE 354 Communication Systems Experiment : 3 Semester: 2016 Spring Group No 1 2 3 4 Student No Group Information Date Student Name Lab. Instructor s Notes Student Signature SAMPLING & TIME DIVISION MULTIPLEX (TDM) PROCEDURE 1. Build the natural sampling system in Figure 4.2 (see Experiment Sheet). In order to generate the switching function in the Figure 4.2, use the TWIN PULSE GENERATOR and DUAL ANALOG SWITCH modules. Take the sampling frequency as 8.333 khz. You can use AUDIO OSCILLATOR module as a message signal with a variable operation frequency. Select an appropriate message frequency to obey Nyquist Criteria! CHANNEL 1 (Message signal) CHANNEL 2 (Sampled signal) Message Signal at. Hz. Sampling Frequency at 8.333 khz. 8/12
Comment: What are the maximum operation frequencies of the message signal to obey Nyquist Critea? Explain, briefly. PROCEDURE 2. Recover the message signal by using the LPF output of the HEADPHONE AMPLIFIER module which has the frequency response of a 3 khz LPF. Select an appropriate message frequency to obey Nyquist Criteria! CHANNEL 1 (Message signal) CHANNEL 2 (Recovered signal) Message Signal at. Hz. Sampling Frequency at 8.333 khz. 9/12
PROCEDURE 3-4. Answer the following questions Comment: What do you observe, when you change the pulse width of the TWIN PULSE GENERATOR? Comment: What do you observe, when you change the message signal frequency? PROCEDURE 5-6. To realize flat top sampling, use the INTEGRATE & DUMP module. Take the sampling frequency as 8.333 khz. You can use AUDIO OSCILLATOR module as a message signal with a variable operation frequency. CHANNEL 1 (Message signal) CHANNEL 2 (Sampled signal) Flat Top Sampling Message Signal sine at. Hz. 10/12
Message Recovery by Tunable LPF module! CHANNEL 1 (Message signal) CHANNEL 2 (Recovered signal) Message Signal at. Hz. 11/12
PROCEDURE 7-8. Use 2 khz sinusoidal signals to observe a 2-channel TDM signal on the oscilloscope and draw the waveform on a scope sheet. Take the sampling frequency as 8.333 khz. 2-Channel TDM Signal CHANNEL 1 (Message signal) CHANNEL 2 (Sampled signal) Flat Top Sampling Message Signal at. Hz. Comment: What do you observe, when you change the time and the width of the samples of the input signals? 12/12