LECTURE 01.pptx Digital electronics for engineers

Summaries

1. Digital transmission of analogue signals LECTURE 1

2. Digital transmission of analogue signals : Sampling theory, spectrum of a chopper sampler, sampling theorem, ideal sampling and reconstruction, practical sampling and aliasing . 2

3. Background As we know, a computer needs to communicate. The information stored in computers are in binary format (0s and 1s ). But the devices are connected only by means of electrical connection. Hence , we must require conversion of binary data into analog signal prior to transmission through the electrical wires . This type of conversion or mapping is called as Digital Modulation . For example, Binary 0 by 0 V and Binary 1 by +5 V 3

4. INTRODUCTION Analog pulse-modulation systems rely on the sampling process to maintain continuous amplitude representation of the message signal . In contrast, digital pulse-modulation systems use not only the sampling process but also the quantization process , which is non-reversible . Quantization provides a representation of the message signal that is discrete in both time and amplitude . In so doing, digital pulse modulation makes it possible to exploit the full power of digital signal-processing techniques . 4

5. Sampling Process S ampling a signal in the time domain has the effect of making the spectrum of the signal periodic in the frequency domain. The sampling process is usually, but not exclusively, described in the time domain . S ampling process is an operation that is basic to digital signal processing and digital communications . Through use of the sampling process, an analog signal is converted into a corresponding sequence of samples that are usually spaced uniformly in time . 5

6. Clearly, it is necessary that we choose the sampling rate properly , so that the sequence of samples uniquely defines the original analog signal. This is the essence of the sampling theorem. 6

7. INSTANTANEOUS SAMPLING AND FREQUENCY-DOMAIN CONSEQUENCES Consider an arbitrary signal of finite energy, which is specified for all time t . A segment of the signal is shown in the figure. 7

8. Suppose that we sample the signal instantaneously and at a uniform rate, once every seconds. Consequently, we obtain an infinite sequence of samples spaced seconds apart and denoted by where n takes on all possible integer values, both positive and negative. We refer to as the sampling period or sampling interval and to its reciprocal as the sampling rate . This ideal form of sampling is called instantaneous sampling . 8

9. Let denote the signal obtained by individually weighting the elements of a periodic sequence of Dirac delta functions spaced seconds apart by the sequence of numbers as shown by (see in figure) 9 is referred to as the instantaneously sampled signal and represents a delta function positioned at time (1)

10. Fourier Transforms of Periodic Signals It is well known that by using the Fourier series , a periodic signal can be represented as a sum of complex exponentials . Also, in a limiting sense, Fourier transforms can be defined for complex exponentials , as demonstrated in Eqs . below 10 for a complex exponential function of frequency Equation (2) states that the complex exponential function is transformed in the frequency domain into a delta function occurring at (2) Complex Exponential Function

11. Sinusoidal Functions We first use Euler’s formula to write 11 Therefore, using Eq. (2), we find that the cosine function is represented by the Fourier-transform pair (3) (4)

12. 12 In other words, the spectrum of the cosine function consists of a pair of delta functions occurring at each of which is weighted by the factor as shown in Fig. below. FIGURE: ( a ) Cosine function. ( b ) Spectrum.

13. Similarly, we may show that the sine function is represented by the Fourier transform pair 13 which is illustrated in the Fig. below FIGURE: ( a ) Sine function. ( b ) Spectrum.

14. SAMPLING THEOREM The link between an analog waveform and its sampled version is provided by what is known as the sampling process . This process can be implemented in several ways, the most popular being the sample-and-hold operation . In this operation, a switch and storage mechanism (such as a transistor and a capacitor, or a shutter and a filmstrip ) form a sequence of samples of the continuous input waveform . The output of the sampling process is called pulse amplitude modulation (PAM) . 14

15. The analog waveform can be approximately retrieved from a PAM waveform by simple low-pass filtering . A bandlimited signal having no spectral components above f m hertz can be determined uniquely by values sampled at uniform intervals of 15 This particular statement is also known as the uniform sampling theorem . Stated another way, the upper limit on can be expressed in terms of the sampling rate , denoted The restriction, stated in terms of the sampling rate, is known as the Nyquist criterion .

16. The statement is 16 The sampling rate is also called the Nyquist rate . The Nyquist criterion is a theoretically sufficient condition to allow an analog signal to be reconstructed completely from a set of uniformly spaced discrete-time samples .

17. 17 SAMPLING Instantaneous Sampling: Suppose we sample an arbitrary signal m ( t ) instantaneously and at a uniform rate , once every [figure (a )]. Then we obtain an infinite sequence of samples where n takes on all possible integer values. This ideal form of sampling is called instantaneous sampling

18. Ideal Sampled Signal : Multiplication of by a unit impulse train [ Fig. (b)] yields 18 The signal [Fig. (c)] is referred to as the ideal sampled signal.

19. 19 Natural Sampling: Although instantaneous sampling is a convenient model, a more practical way of sampling a band-limited analog signal is performed by high-speed switching circuits . An equivalent circuit employing a mechanical switch and the resulting sampled signal are shown in Fig. (a ) and (b ), respectively Practical Sampling :

20. The sampled signal can be written as 20 where is the periodic train of rectangular pulses with period and each rectangular pulse in has width and unit amplitude. The sampling here is termed natural sampling , since the top of each pulse in retains the shape of its corresponding analog segment during the pulse interval.

21. Flat-Top Sampling: The simplest and thus most popular practical sampling method is actually performed by a functional block termed the sample-and-hold (S/H) circuit [ Fig. (a )]. This circuit produces a flat-top sampled signal [ Fig . (b )]. 21

22. ALIASING In this derivation of the sampling theorem, the assumption is that the signal is strictly band-limited . In practice, there is no information-bearing signal of physical origin is strictly band-limited , with the result that some degree of undersampling is always encountered. Consequently, aliasing is produced by the sampling process . 22

23. Aliasing refers to the phenomenon of a high-frequency component in the spectrum of the signal seemingly taking on the identity of a lower frequency in the spectrum of its sampled version , as illustrated in Fig. below. The aliased spectrum shown by the solid curve in Fig. (b) pertains to an “undersampled” version of the message signal represented by the spectrum of Fig. (a ) 23

24. To combat the effects of aliasing in practice, we may use two corrective measures: Prior to sampling, a low-pass anti-alias filter is used to attenuate those high-frequency components of a message signal that are not essential to the information being conveyed by the signal . The filtered signal is sampled at a rate slightly higher than the Nyquist rate . 24

25. The use of a sampling rate higher than the Nyquist rate also has the beneficial effect of easing the design of the synthesis filter used to recover the original signal from its sampled version . Consider the example of a message signal that has been anti-alias ( low-pass) filtered , resulting in the spectrum shown in Fig. (a ). The corresponding spectrum of the instantaneously sampled version of the signal is shown in Fig. (b ), assuming a sampling rate higher than the Nyquist rate . 25

26. we now readily see that the design of a physically realizable reconstruction filter aimed at recovering the original signal from its uniformly sampled version may be achieved as follows ( see Fig . (c )): The reconstruction filter is of a low-pass kind with a passband extending from to which is itself determined by the anti-alias filter . The filter has a non-zero transition band extending (for positive frequencies) from to where is the sampling rate 26