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

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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. Quanitization
Syed M. Zafi S. Shah

16. 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

17. 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

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

19. 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

20. 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

21. 3-bit ADC Quant. error 8-bit ADC Quant. error
Observe the 10e-3 in the 4rth figure
8-bit ADC
Quant. error
Syed M. Zafi S. Shah

22. 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

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

24. 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

25. 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

26. لڳ ڀڳ ڪُل اوقاتي Near Continuous

27. T=0.1

28. T=?

29. T=?

30. 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

31. 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

32. 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