# Analog to digital conversion

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Analog to digital conversion
Syed M. Zafi S. Shah احسان احمد عرساڻي Lecture 3,4 Syed M. Zafi S. Shah

In today’s class Analog to digital conversion Sampling quantization
Sampling theorem, aliasing quantization Quantization noise Syed M. Zafi S. Shah

Need for A/D conversion
We know by now the benefits of digital signals and systems But most signals of practical interest are still analog Voice, Video RADAR signals Biological signals etc So in order to utilize those benefits, we need to convert our analog signals into digital This process is called A/D conversion Syed M. Zafi S. Shah

Three step process Analog to Digital conversion is really a three step process involving Sampling Conversion from continuous-time, continuous valued signal to discrete-time, continuous-valued signal Quantization Conversion from discrete-time, continuous valued signal to discrete-time, discrete-valued signal Coding Conversion from a discrete-time, discrete-valued signal to an efficient digital data format Represent as bit? Syed M. Zafi S. Shah

Analog signal Binary bits CT-CV DT-CV DT-DV DT-DV
SAMPLING QUANTIZATION CODING CT-CV DT-CV DT-DV DT-DV Arbitrarily, I’ve chosen Differential PCM…. Can you re-create these graphs? Syed M. Zafi S. Shah

Sampling A continuous-time signal has some value ‘defined’ at ‘every’ time instant So it has infinite number of sample points sample every 1 sec So, the question is ‘What should be the duration for taking a sample?’ Should it be the 2nd, 3rd or 4th figure?, Which one would you prefer? What can be the advantages of taking more samples? What can be its dis-advantages? What can be the advantages of taking less samples? What can be its dis-advantages? sample every 0.1 sec sample every 1 μsec Syed M. Zafi S. Shah

So we have to take some finite number of points
Aliasing: It is impossible to digitize an infinite number of points because infinite points would require infinite amount of memory and infinite amount of processing power So we have to take some finite number of points Sampling can solve such a problem by taking samples at the fixed time interval If an analog signal is not appropriately sampled, aliasing will occur, where a discrete-time signal may be a representation (alias) of multiple continuous-time signals Syed M. Zafi S. Shah

Shannon’s sampling theorem
The sampling theorem guarantees that an analogue signal can be in theory perfectly recovered as long as the sampling rate is at least twice as large as the highest-frequency component of the analogue signal to be sampled A signal with no frequency component above a certain maximum frequency is known as a band-limited signal (in our case we want to have a signal band-limited to ½ Fs) Some times higher frequency components are added to the analog signal (practical signals are not band-limited) In order to keep analog signal band-limited, we need a filter, usually a low pass that stops all frequencies above ½ Fs. This is called an ‘Anti-Aliasing’ filter Students will already know about the fact that signals are not band-limited in the DSP practical Syed M. Zafi S. Shah

In order to sample a voice signal containing frequencies up to 4 KHz, we need a sampling rate of 2*4000 = 8000 samples/second Similarly for sampling of sound with frequencies up to 20 KHz, we need a sampling frequency of 2*20000 = samples/second What is the sampling rate for CDs? Isn’t it more than the one we just calculated? Syed M. Zafi S. Shah

The maximum frequency component is x(t) is
Example 1: For the following analog signal, find the Nyquist sampling rate, also determine the digital signal frequency and the digital signal The maximum frequency component is x(t) is Therefore according to Nyquist, we need a sampling rate of The digital signal would have a frequency The digital signal can be represented as Syed M. Zafi S. Shah

Anti-aliasing filters
Anti-aliasing filters are analog filters as they process the signal before it is sampled. In most cases, they are also low-pass filters unless band-pass sampling techniques are used The ideal filter has a flat pass-band and the cut-off is very sharp, since the cut-off frequency of this filter is half of that of the sampling frequency, the resulting replicated spectrum of the sampled signal do not overlap each other. Thus no aliasing occurs Syed M. Zafi S. Shah

Practical low-pass filters cannot achieve the ideal characteristics
Practical low-pass filters cannot achieve the ideal characteristics. What can be the implications? Firstly, this would mean that we have to sample the filtered signals at a rate that is higher than the Nyquist rate to compensate for the transition band of the filter That’s why the sampling rate of a CD is 44.1 KHz, much higher than the 40 KHz we calculated Go through the assignment… it has some reading task along with some problems Syed M. Zafi S. Shah

Example 2: Find the Nyquist’s rate for the following signal
This composite signal comprises three frequencies f1 = 25 Hz, f2 = 150 Hz, f3 = 50 Hz So, according to Nyquist we need to sample at 300 Hz However, for the ‘sine’ term, the sampled signal has values sin(πn), meaning the samples are taken at the ‘zero crossings’, so the sine term is not counted in the process Therefore, a solution is to sample at higher than twice the max. freq component Syed M. Zafi S. Shah

Quantization Now that we have converted the continuous-time, continuous-valued signal into a discrete-time, continuous-valued signal, we STILL need to make it discrete valued This is where Quantization comes into picture “The process of converting analog voltage with infinite precision to finite precision” For e.g. if a digital processor has a 3-bit word, the amplitudes of the signal can be segmented into 8 levels Syed M. Zafi S. Shah

Quanitization Syed M. Zafi S. Shah

General rules for Quantization
Important properties of the quantizer include Number of quantization levels Quantization resolution Note the minimum & maximum amplitude of the input signal Ymin & Ymax Two important properties of quantizer include the number of quantization levels and quantization resolution. In order to calculate them we need to know about the following… Syed M. Zafi S. Shah

Note the word-length of DSP Number of levels of quantizer is equal to
m-bits Number of levels of quantizer is equal to L = 2m The resolution of the quantizer is given as Resolution of a quantizer is the distance between two successive quantization levels More quantization levels, better resolution! Whats the downside of more quantization levels? Syed M. Zafi S. Shah

m = 4, L = 16 Ymin = 0 Ymax = 1 ∆ = 1/15 = Syed M. Zafi S. Shah

Quantization error The error caused by representing a continuous-valued signal(infinite set) by a finite set of discrete-valued levels Suppose a quantizer operation given by Q(.) is performed on continuous-valued samples x[n] is given by Q(x[n]), then the quantization error is given by If x(n) lies between two quantization levels, it will either be rounded up or truncated Rounding replaces x(n) by the value of the nearest quantization level. Truncation replaces x(n) by the value of the level below it Syed M. Zafi S. Shah

Lets consider the signal , which is to be quantized.
In the figure (previous slide), we saw that there was a difference between the original signal and the quantized signal. This is the error produced while quantization It can be reduced, however, if the number of quantization levels is increased as illustrated on next slide Syed M. Zafi S. Shah

Observe the 10e-3 in the 4rth figure 8-bit ADC Quant. error Syed M. Zafi S. Shah

Signal-to-Quantization-noise ratio
Provides the ratio of the signal power to the quantization noise (or quanitization error) Mathematically, where Px = ¨Power of the signal ‘x’ (before quantization) Pq = ¨Power of the error signal ‘xq’ Syed M. Zafi S. Shah

The sampled signal in the FD جُزايل سگنل ڪثرتي ميدان ۾
If Then

The Ideal Sampling Theorem خيالي جُزڪاري نظريئڙو
Also called Uniform Sampling Theorem Nyquist Theorem If x(t) is band limited with no components at frequencies greater than fh Hz then it is completely specified by samples taken at the uniform rate fs>2fh 2fh is called the Nyquist rate

The Sampling Rate جُزڪاريءَ جي شرح
The no. of samples per second هڪ سيڪنڊ ۾ جُزن جو تعداد If the sampling rate: fs>2fh , it is called Over-sampling fs=2fh , it is called Critical-sampling fs<2fh , it is called Under-sampling

لڳ ڀڳ ڪُل اوقاتي Near Continuous

T=0.1

T=?

T=?

Anti-aliasing filters
Anti-aliasing filters are analog filters as they process the signal before it is sampled. In most cases, they are also low-pass filters unless band-pass sampling techniques are used The ideal filter has a flat pass-band and the cut-off is very sharp, since the cut-off frequency of this filter is half of that of the sampling frequency, the resulting replicated spectrum of the sampled signal do not overlap each other. Thus no aliasing occurs Syed M. Zafi S. Shah

Practical low-pass filters cannot achieve the ideal characteristics
Practical low-pass filters cannot achieve the ideal characteristics. What can be the implications? Firstly, this would mean that we have to sample the filtered signals at a rate that is higher than the Nyquist rate to compensate for the transition band of the filter That’s why the sampling rate of a CD is 44.1 KHz, much higher than the 40 KHz we calculated Go through the assignment… it has some reading task along with some problems Syed M. Zafi S. Shah

Example 2: Find the Nyquist’s rate for the following signal
This composite signal comprises three frequencies f1 = 25 Hz, f2 = 150 Hz, f3 = 50 Hz So, according to Nyquist we need to sample at 300 Hz However, for the ‘sine’ term, the sampled signal has values sin(πn), meaning the samples are taken at the ‘zero crossings’, so the sine term is not counted in the process Therefore, a solution is to sample at higher than twice the max. freq component Syed M. Zafi S. Shah

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