ed 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. Source Coding and Decoding: Making it Digital 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(A) 2. Determine the Nyquist sampling rate for the following signals (a) x(1)-sin(40007t1) (Q4.1) (Q4.2) 3. Consider the signal y(t)=x,(t)*x2(t) (Q4.3) where 4(f)=0, / > 1000 Hz (Q4.4) X2(f)=0, f|>2000 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? Figure Q4.1 The input -f1+Af Chapter Four 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) 