Congrats! The end of a (rather lengthy) chapter. To recap, and I’ll be brief, we learned in detail about two parts of the source coder, the sampler and the quantizer. First, we learned about three types of samplers and saw the coolest of results: as long as you sample at (at least) two times the maximum frequency, you can get your original signal back from your samples. Then we learned about quantizers, a fancy word for an
“amplitude changer”—it maps the input amplitude to one of N allowed output ampli- tudes. You also saw two ways to build a quantizer that minimize the average error between input and output. Not only that, but you saw the quantizer most commonly used in telephone communications.
Then we decided to get adventurous, and we put the sampler and quantizer to- gether and built a source coder called a PCM. Next, we considered a different source coder called the predictive coder. It looked a lot like the PCM, the only difference being that, before you got to the quantizer, you removed a predicted value from your sample. We talked at length about two different types of predictive coders, the DM and the DPCM, and finally, you came here to the summary, where we wrapped it all up.
If you feel you want more on this material, and be assured there are books and books on this stuff, have a look at the references.
Problems
1. If you know that the signal x(t) is completely characterized by the samples taken at a rate of 5,000 Hz, what (if anything) can you say about X(f)?
2. Determine the Nyquist sampling rate for the following signals
(a)
( )
(
)
t
t
t
x
π
π
4000
sin
=
(Q4.1) (b)( )
(
2 2)
24000
sin
t
t
t
x
π
π
=
(Q4.2)3. Consider the signal
( ) ( ) ( )t
x
t
x
t
y
=
1∗
2 (Q4.3) where( )
10,
1000
X
f
=
f
>
Hz
(Q4.4)( )
20,
2000
X
f
=
f
>
Hz
(Q4.5)What is the minimum sampling period that ensures that y(t) is completely recov- erable from its samples?
4. Assume the signal x(t) has the frequency representation shown in Figure Q4.1. (a) What does the output of an ideal sampler look like in the frequency
domain?
(b) What is the minimum sampling rate that I can use and still recover my signal from its samples?
X(f)
f –f1–∆f –f1 –f1+∆f f1–∆f f1 f1+∆f
5. Consider zero-order hold sampling.
(a) If the input x(t) has a Fourier transform shown in Figure Q4.2, what does the output waveform look like (1) in the frequency domain and (2) in the time domain? Assume sampling at the Nyquist rate. (b) If the input x(t) has a Fourier transform shown in Figure Q4.2, what
does the output waveform look like (1) in the frequency domain and (2) in the time domain? Assume sampling at TWICE the Nyquist rate.
x(f)
–fm fm
f
Figure Q4.2 The input
6. Plot the output of a sampler in frequency given:
• The input signal has maximum frequency 5,300 Hz. • Ideal sampling is used.
• Sampling is at a rate of 5,300 Hz.
• The input signal in the frequency domain is triangular (i.e., it is a maximum at 0 Hz and degrades to 0 linearly as frequency increases to 5,300 Hz (and to –5,300 Hz).
7. Consider a quantizer with an input described in Figure Q4.3.
(a) Draw a quantizer with 7 levels. Make it mid-tread, let it have –3 as its smallest output value, and make sure that the step size is 1.
(b) Evaluate the mse of your quantizer given the input. (c) Evaluate the SQNR.
(d) If the input to the quantizer has an amplitude with a probability distribution uniform between –3.5 and +3.5, what is the SQNR of the quantizer?
1/4 1/6 p(x)
–3 2
x
Figure Q4.3 The input pdf
8. Find out how many levels a quantizer must use to achieve an SQNR greater than 30 dB given:
• The incoming audio signal is sampled at its Nyquist rate of 8,000 samples/ sec.
• The amplitudes output from the sampler have a uniform probability distri- bution function.
• A uniform quantizer is used.
9. Determine the optimal compression characteristic for the input x whose prob- ability density function is provided in Figure Q4.4.
10. (a) Plot the µ =10 compressor characteristic given that the input values are in the range [–2.5,2.5].
(b) Plot the corresponding expander.
1/6 1/12 p (x) –4 –2 2 x 4
11. Evaluate the symbol rate and the bit rate of the PCM system described by the following:
• The sampling rate is 5,300 Hz • The quantizer is an 8-level quantizer. 12. A computer sends:
• 100 letters every 4 seconds • 8 bits to represent each letter
• the bits enter a special coding device that takes in a set of bits and puts out one of 32 possible symbols.
What is the bit rate and what is the symbol rate out of the special coding device? 13. Over the time 0 s to 2 s, determine (1) the input to the DM, (2) the output of the DM, and (3) the times of granular and overload noise given:
• The input to the DM is x(t) = t2
• The sampler offers 10 samples/second • The step size of the DM is 0.1 V 14. (a) Draw the output of the DM given:
• The input corresponds to x(t)=1.1t + 0.05 • The input is sampled at times t = 0,1,2,3,4 • The step size is 1 and a= 1.
(b) Repeat (a), this time using a= 0.5.
15. A two-tap predictive filter is being designed to operate in a DPCM system. The predictor is of the form
2 2 1 1 − + − = n n P n a x a x x (Q4.6)
(a) Provide an explicit equation for the optimal selection of a1 and a2 (in terms of autocorrelation functions) which minimizes the mean squared prediction error.
(b) Provide a general equation for the mean squared prediction error using the values determined in (a).
(c) Determine the values of the predictor taps in (a) and the prediction error in (b) given: