ECE 2111 Signals and Systems Spring 2012, UMD Experiment 9: Sampling Objective: In this experiment the properties and limitations of the sampling theorem are investigated. A specific sampling circuit will be constructed and tested using a variety of input signals and sampling signals. Equipment and Material: - 3 Function Generators - Oscilloscope - LF 398 sample-and-hold chip - MF4CN-100 fourth order Butterworth low-pass filter - 4-10 μf capacitors ( 50 V) BACKGROUND 1. - Sampled Signals Analog signals, which are the most familiar type of signal, are continuous functions of time in the sense that their amplitudes are defined explicitly for every instant of time. However, there is another important class of signals, usually referred to as sampled signals, for which the amplitude is defined (non-zero) only for a certain discrete instant of time. Fig. 4.1 displays an example of both the analog and a sampled signal. Sampled signals are used in pulse-modulation communication systems, in sampled data control systems, and when digital computers are used as part of an analog system. Fig. 4.1 Examples of: (a) an analog signal; (b) a sampled signal The process of generating sampled signals, sometimes called pulse-amplitude modulation is illustrated in Fig. 4.2. The analog input signal, x i (t), is multiplied by the uniform pulse train. x s (t), and the resulting output signal, x o (t), is non-zero only when x i (t) and x s (t) are both non-zero. The analog signal, x i (t), is said to have been sampled by the sampling signal, x s (t). An equivalent method of describing the sampling process is the single-pole, single-throw switch shown in Fig. 4.3. 1
Fig.4.2. Generation of Sampled Signals Fig.4.3 Representation of the Sampling Process 2.-Sampling Theorem Sampled signals such as x o (t) in Fig. 4.2(d) are useful only if they contain the same information as the original signal, x i (t), as shown in Fig.4.2(b). That is to say, x i (t) must be recoverable from x o (t). The conditions under which such a recovery of the original signal constitute a statement of the sampling theorem. Briefly these conditions are: 1. - The original signal x i (t) must be a band-width limited function (i.e., have no frequency components outside the frequency interval [ -f b to + f b ], and 2. - The frequency of the sampling signal, x s (t), must be greater than 2f b. 2
Fig.4.4. Illustration of the sampling theorem Consider the signal x i (t) shown in Fig. 4.4 (a), which has a band-width limited spectrum, also shown in Fig. 4.4 (a). This signal is sampled by the uniform impulse train, x δs (t), shown in Fig. 4.4 (b). The spectrum of x δs (t), is itself a uniform impulse train, X δs (f), in the frequency domain, as is shown in Fig. 4.4 (b) (from Fourier analysis). The output signal, x δo (t), is easily found in the time domain, as is illustrated in Fig. 4.4(c). The output spectrum, X δs (f), is found by the convolution of X i (f) and X δs (f). This is due to the fact (from Fourier analysis) that multiplication in the time domain is equivalent to convolution in the frequency domain. Thus When the convolution indicated by eq. (1) is carried out, the output, X δo (f), shown in Fig. 4.4(c) results. Inspection of the output spectrum, X δo (f), shows that the spectrum of the original signal is reproduced symmetrically about each frequency harmonic of the sampling signal. The original spectrum can be recovered from X δo (f), through low-pass filtering, as shown in Fig. 4.5(a). This is true as long as neighboring replicas of the input spectrum do not overlap. In order to avoid the following conditions must be met: 1. - The input, X i (f), must have no frequency components outside the frequency interval, - f b to + f b, and 2. - The sampling frequency, l/t s, must be greater than or equal to twice the band-limit of the input signal 2f b. 3
These two conditions form a restatement of the sampling theorem. A complete block diagram of an ideal sampling system is shown in Fig. 4.5 (b). Fig.4.5. Recovery of original signal from sampled signal 3. - Effect of Finite Duration Sampling Pulses A sampling composed of a set of infinitely short pulses cannot be realized in practice, so the sampling operation just described must be modified to take the finite duration of the sampling pulses into account. Consider the sampling pulse train shown in Fig. 4.6 (a). The spectrum of this pulse train is shown in Fig. 4.6(b). Fig.4.6. Finite width pulse amplitude spectrum Note: The value of τ p should be evenly divisible by Ts. When the spectrum shown in Fig.4.6 (b) is convolved with X i (f) shown in Fig.4.4 (a), the output spectrum X o (f), shown in Fig.4.7 results. 4
Fig.4.7. Amplitude spectra of X 0 (f): in a system with a finite width sampling pulse, and the amplitude transfer characteristics of the low-pass filter used to recover the original signal from the sampled signal. The original signal, x i (t), can still be recovered from the new output signal, x o (t) by low-pass filtering as an examination of Fig. 4.7 reveals. The effect of the finite width of the sampling pulse is to reduce the gain of the sampling system. In the ideal case Whereas the finite-width sampling pulse gives Note: The gain of the low-pass filter is assumed to be one 5
4. - Practical Sampling Circuit The sampling and reconstruction circuit used in this experiment is displayed in Fig.4.8. Fig.4.8. Practical sampling and reconstruction circuit The LF398 in Fig.4.8 is a sample-and-hold chip. With a short circuit from pin 6 to ground, the hold is disabled and the chip becomes a sampler. The MF4cn-100 is a fourth-order, switched-capacitor, Butterworth lowpass filter that will reconstruct the sampled signal. The input signal X 100c (t) is a TTL (0-5 V) square wave. The cut off frequency of the filter can be selected by choosing the frequency of X 100c (t), which will be 100 times larger than the cut off frequency of the filter. The maximum cut-off frequency of MF4cn-100 is 10 KHz. (refer data sheet). The four 10µF capacitors will cut out the noise on the power line. The signals related to the circuit in Fig.4.8 are x i (t) : input signal x s (t) : sampling signal x o (t) : sampled signal with finite pulse width x100c(t) : cut off frequency control signal (will be 100 times larger than the cut off frequency of the filter) x LP (t) : Reconstructed signal 6
PROCEDURES 1.- Prepare: Circuit of Fig.4.8: x i (t) = 500mVpp, 10kHz, sine wave, no DC component (no offset on the function generator); x s (t) = 5Vpp of amplitude and 2.5V of offset, 40kHz, 50% duty cycle, square wave; and x100c(t) = 5Vpp of amplitude and 2.5V of offset, square wave, 50% duty cycle (duty cycle=one pulse width/one cycle), 1MHz (100 times larger than the cut off) 2.- Prepare: Circuit of Fig. 4.8: x i (t) = 500mVpp, 10kHz, sine wave, no DC component; x s (t) = 5Vpp of amplitude and 2.5V of offset, 15kHz, 50% duty cycle, square wave; and x100c(t) = 5Vpp of amplitude and 2.5V of offset, 50% duty cycle, square wave, 1MHz (100 times larger than the cut off) 3.- Prepare: Circuit of Fig. 4.8: xi(t) = 500mVpp, 10kHz, sine wave, no DC component; x s (t) = waveform of Fig.4.9, 20% duty cycle ; x100c(t) = 5Vpp of amplitude and 2.5V of offset, 50% duty cycle, square wave, 1MHz (100 times larger than the cut off) Fig. 4.9 Sampling pulse train 4.- Prepare: Circuit of Fig. 4.8: x s (t) = 5Vpp of amplitude and 2.5V of offset, 40kHz, 50 % duty cycle, square wave x100c(t) = 5Vpp of amplitude and 2.5V of offset, 50% duty cycle, square wave, 1MHz (100 times larger than the cut off) a) xi(t) = 500mVpp, 2kHz square wave, no DC component 7
b) xi(t) = 500mVpp, 5kHz square wave, no DC component c) xi(t) = 500mVpp, 2kHz triangular wave, no DC component d) xi(t) = 500mVpp, 5kHz triangular wave, no DC component Report Requirements: Individual lab report Cover page attached (available on the website) The lab report contains the following: 1. Statement of the Problem: Define the problem and goals of the experiment. 2. Results: a. Screenshots of oscilloscopes b. Comments 3. Exercises: Attached 4. Conclusions: Give comments to the lab. Explain discrepancies between theory and practical results of experiments, if any. Exercises: 1a). For a signal x(t) = 500mVpp, 10kHz, sine wave, no DC component, what is the minimum required sampling frequency? 1b). If the above signal x(t) is sampled with a 18KHz frequency signal, will aliasing occur? Yes/ No (Circle one) 8
Why? Or why not? 2) Sketch the filter to reconstruct the original signal (bottom) from the sampled signal (top). 9