FROM ANALOGUE TO DIGITAL

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1 SIGNALS AND SYSTEMS: PAPER 3C1 HANDOUT 7. Dr David Corrigan 1. Electronic and Electrical Engineering Dept. corrigad FROM ANALOGUE TO DIGITAL To digitize signals it is necessary to define them by samples of discrete amplitude, taken at discrete instants in time. Each sample amplitude is quantized so that its voltage is represented by a number which has a minimum (typically 0) and a maximum value (depends on the number of bits allocated for representing the number 8 bits = 255 max, 16bits = max). Much signal processing today is done in the digital domain. Digital signals can be manipulated by computers which are intrinsically very flexible i.e. programmable. Because digital signals can be easily manipulated one can make digital systems which implement much more complicated processes than would be possible using analogue techniques. Digital signal processing is encountered in everyday devices like mobiles, CD,.mp3 Players DVD Players (Digital Video Disc.), soundblaster cards for the PC, web-cams, network cards etc. Even cars, microwaves... For digital processing to be any use we must be confident that representing a signal by a set of discrete samples still retains all the information of the original analogue signal. We must then be also confident that it is possible to reconstruct the analogue signal from the samples after the samples have been processed. We ll first look at the effects of sampling and then look at techniques for reconstructing the analogue signal from these samples. 1 This handout is based on the set of notes produced by Prof. Anil Kokaram 3C1 Signals and Systems 1

2 1 SAMPLING 1 Sampling C1 Signals and Systems 2

3 1.1 A note about representing spectra 1 SAMPLING 1.1 A note about representing spectra We will be deriving limits on sampling using spectal analysis. To do this we will be drawing a bunch of representations for signal spectra. Most naturally occuring (and interesting) signals have most of their spectral energy concentrated around the lower frequencies. In fact most natural signals (speech, music, pictures of natural scenes like landscapes) have spectra with energy that falls off with 1/f where f is frequency Frequency (Hz) Frequency (Hz) Frequency (Hz) So to visualise what is happening to signals in terms of frequency content, we tend to use pictures like below. Do not be alarmed by these pictures, they are just representations of signal spectra to allow us to visualise what would happen to the frequency content. In fact the most important thing about these pictures is that they allow us to visualise the bandwidth of signals without having to worry about a specific signal. 3C1 Signals and Systems 3

4 1.2 Sampling (Quantitative analysis) 1 SAMPLING 1.2 Sampling (Quantitative analysis) The goal is to develop an idealised mathematical model that describes the sampling process. Recall, the sifting property of the delta function x(t)δ(t u)dt = x(u) (1) In effect, the sifting property describes the sampling of x(t) at time t = u. Therefore we can model sampling over the entire signal by applying the sifting property to a series of different impulse functions located at regular intervals along the time axis. This sequence of impulses is called an impulse train. The impulse train p(t) selects the values of x(t) only at the desired sampling instants. We wish to examine the spectrum of the sampled signal and compare it to the spectrum of the original analogue signal. 3C1 Signals and Systems 4

5 1.2 Sampling (Quantitative analysis) 1 SAMPLING x(t) input signal Ideal sampler. p(t) sampling impulses x S (t) sampled signal Ideal Sampler x S (t) = x(t) p(t) (2) The sampling pulses p(t) are a periodic train of impulses of unit mean value and period T secs: p(t) = T δ(t mt ) (3) m= 3C1 Signals and Systems 5

6 1.2 Sampling (Quantitative analysis) 1 SAMPLING In order to make it easier to calculate the spectrum of x S (t), we use Fourier series analysis with ω 0 = 2π/T to express p(t) instead as a sum of its Fourier components: p(t) = a k e jkω 0t where a k = 1 T k= T/2 = e jkω 0. 0 T δ(t) e jkω 0t dt T/2 T/2 = 1 for all n. p(t) = n= e jnω 0t T/2 δ(t) dt 3C1 Signals and Systems 6

7 1.2 Sampling (Quantitative analysis) 1 SAMPLING An interesting point to note here is that, since the impulse train is a periodic function, the frequency spectrum of the impulse train is also a impulse train. In the frequency domain the interval between impulses is equal to the fundamental frequency which is the sampling frequency. The value of the fourier coefficients are uniform. Now we want to find X S (ω) so: x S (t) = x(t) p(t) = (4) Note that the sampling frequency, f S = 1 T Hz = ω 0 2π rad/sec. Procced to find X S (ω) by taking Fourier transforms and using the frequency shift theorem: HENCE X s (ω) = F(x s (t)) = = = = X S (ω) = k= k= n= x s (t)e jωt dt [ k= [ [ ] x(t)e jkω 0t e jωt dt ] x(t)e jkω0t e jωt dt ] x(t)e j(ω kω0)t dt X(ω nω 0 ) Alternatively we could have derived this result using the frequency shift and linearity properties of the fourier transform. 3C1 Signals and Systems 7

8 1.2 Sampling (Quantitative analysis) 1 SAMPLING Hence the spectrum of a sampled signal is the original spectrum X(ω), repeated at multiples of the sampling freq. See below Spectral representation of sampling. 3C1 Signals and Systems 8

9 1.2 Sampling (Quantitative analysis) 1 SAMPLING If the repeated spectra overlap, ALIASING occurs. (a) No Aliasing (b) Aliasing Signal recovery and the effect of aliasing. x(t) x S (t) Lowpass Filter y(t) recovered signal p(t) Lowpass filter recovers the original signal. If there is no aliasing, the original signal can be recovered perfectly from the sampled version by lowpass filtering, which removes all repeats of the original spectrum. 3C1 Signals and Systems 9

10 1.2 Sampling (Quantitative analysis) 1 SAMPLING If aliasing occurs, perfect recovery is not possible - f S is too low! X( ω+ω s ) X(ω) ω s - ω max X(ω ω s ) ω - ω s 0 ω max ω s As long as ω s ω max > ω max there is no aliasing. Therefore, minimum sampling frequency is when ω s ω max = ω max which implies ω s = 2ω max Same for Hz since ω s = 2πf S. The Sampling Theorem (Nyquist): If a signal has components with a maximum frequency of W Hz, then it is completely defined by samples which occur at intervals of 1/2W sec. The Sampling Theorem (Nyquist): If a signal has components with a maximum frequency of W Hz, then it is completely defined by samples that are created by sampling at 2W Hz. That is to say: if it is required to sample an analogue signal which has a spectrum containing frequency components up to W Hz, then the minimum sampling frequency required is 2W Hz. When this condition is satisfied it is possible to reconstruct the analogue signal exactly from the sampled data. This minimum sampling frequency required to prevent aliasing is called the Nyquist frequency. 3C1 Signals and Systems 10

11 2 QUANTIZATION 2 Quantization C1 Signals and Systems 11

12 2 QUANTIZATION Now we consider the effects of discrete quantization of the amplitude of our time samples. The figure below shows a digital coding system. x(t) Ideal Sampler x S (t) Quantiser x Q (t) Digital coding system. Quantization results in a transfer function similar to that shown in the figure below. A-D + D-A transfer function (ideal linear quantizer). 3C1 Signals and Systems 12

13 2 QUANTIZATION If the quantizer has a small step size and it always rounds to the nearest output value, a given input voltage results in a random quantizing error that has a uniform probability of being anywhere from δv/2 to +δv/2. Hence the quantizing errors may be modelled as additive random noise with a uniform probability density function (p.d.f.). Using these ideas we can estimate the average error caused by quantization and so work out how many bits we would need to represent a particular signal properly. This will not be discussed further here. x S (t) sampled signal quantising error e(t) + x Q (t) quantised signal Quantising noise model. p.d.f. of e(t) 1/δv δv/2 0 δv/2 e(t) Quantising noise probability density function (p.d.f.). 3C1 Signals and Systems 13

14 2 QUANTIZATION Quantising image samples Clockwise from top left: 8bit (256 Grey levels), 5bit (32 Grey Levels), 1bit (2 Grey levels), 2bit (4 Grey Levels) Quantisation. 3C1 Signals and Systems 14

15 3 Anti-aliasing/smoothing filters 3 ANTI-ALIASING/SMOOTHING FILTERS In most digitizing systems, 2 lowpass filters are required, as below. x Input Filter x F Sample & Hold x S A-D Conv. D-A Conv. y S Output Filter y Filters required for A-D and D-A conversion. Input filter (Anti-Aliasing Filter) prevents possible aliasing distortion by eliminating any input frequencies > 1 2 f S. A-D converter converts the sampled signal into a digital signal. This includes quantisation of the sampled signal. D-A converter converts the digital signal back into an analogue signal. Output filter removes all components of the sampled signal spectrum > 1 2 f S, to regenerate the original continuous signal. We want to examine the role of the input filter in the overal system. For now we will assume that the D-A converter is perfect and will ignore the effects of quantisation. Hence, we assume that x S (t) = y S (t). Next fig. shows the filter actions. 3C1 Signals and Systems 15

16 3 ANTI-ALIASING/SMOOTHING FILTERS 3C1 Signals and Systems 16

17 3 ANTI-ALIASING/SMOOTHING FILTERS Let the filters have a passband of bandwidth f P. Realisable filters must also have a transition band of finite width f T between the passband and the stopband. To make f T narrow, the filter complexity (no. of poles and zeroes) must be high. Passband Transition Band Stopband Typical filter response. 3C1 Signals and Systems 17

18 3 ANTI-ALIASING/SMOOTHING FILTERS In both filters, the stopband must start at 1 2 f S to avoid aliasing and unwanted output components. Hencef P + f T 1 2 f S (5) If the required signal bandwidth = W, then f P W for negligible frequency distortion. Let us assume that f T = αf P for a given complexity of filter. Typically α = 0.1 (high complexity) to 1.0 (low complexity). Therefore the minimum sampling frequency f S is given by: f S = 2(f P + f T ) = 2W (1 + α) (6) Therefore if low filter complexity is important, we must choose f S to be significantly greater than 2W (e.g. 4W ). 3C1 Signals and Systems 18

19 3 ANTI-ALIASING/SMOOTHING FILTERS Sampling in images Top left: 4th Order butterworth filter (on rows then columns) for anti-aliasing Top right: No anti-aliasing filter Bottom: Analogue lenna 2 2 In fact, that s a lie, this picture is also digital, but the sampling rate used was 3 times higher than the top two images and the Nyquist limit is satisfied (in this case), even though an anti-aliasing filter was not used. 3C1 Signals and Systems 19

20 4 A PRACTICAL SYSTEM 4 A Practical system Although sampling signals using impulse trains is mathematically attractive it is not practical to implement. A more practical alternative is called a sample and hold filter. The idea is to sample the value at each interval and hold the value signal constant until the next sample occurs. This is illustrated below D-A converter hold action. Ignoring quantisation errors, the A-D + D-A system is equivalent to an ideal sampler followed by a filter which extends each sampling impulse into a pulse of width T = 1/f S, as below. 3C1 Signals and Systems 20

21 4 A PRACTICAL SYSTEM x(t) x S (t) Hold Filter h(t) = y S (t) p(t) Equivalent circuit for D-A hold action. If the filter impulse response is given by: then h(t) = 1 for 0 t < T, = 0 elsewhere, H(ω) = T ( ) ωt sinc = T 2 (π sinc f ) (7) f S See below for the effect of this filtering on the spectrum Y S (ω) of the D-A output. It is possible to modify the Output Filter response as shown so as to correct for the hold filter and obtain a flat response overall. 3C1 Signals and Systems 21

22 4 A PRACTICAL SYSTEM Spectra of signals modified by D-A hold action. 3C1 Signals and Systems 22

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