In this project, students are required to design a program to perform analog to digital (A/D)  conversion  and  digital  to  analog  (D/A)  conversion  on  Matlab  or  Pathon. Specifically, given the input signal x(t)=A1cos(2πf1t)+A2sin(2πf2t) (the way to generate the parameters A1, f1, A2, and f2 will be introduced later), 0≤t≤1, you first need to act as a transmitter to perform the A/D conversion via sampling and quantization. Assume that these quantization bits can be sent to the receiver without any error. Then, you act as a receiver and perform the D/A conversion via de-quantization and de-sampling based on your received quantization bits. You will see what is your estimated signal y(t), 0≤t≤1, and what is the error to the original signal, i.e., y(t)-x(t), 0≤t≤1.

Note  that  in  Matlab  and  Pathon,  there  are  functions  to  perform  A/D  and  D/A conversions directly. You are not allowed to call these functions. Please design your code based on the following instructions.

Step 1: Initializing the parameters of input signals. Please generate the parameters A1, f1, A2, and f2 in the input signal x(t)=A1cos(2πf1t)+A2sin(2πf2t), 0≤t≤1, using the first 8 digits of your Student ID. Specifically, A1 is the sum of the first four digits of your ID and A2 is the sum of the last four digits of your ID. Then, convert the last four digits of your ID, which is a decimal number, into a 16-bit binary number (you may do the conversion using the Matlab function “dec2bin” or some online conversion website). For the corresponding 16-bit binary number, f1 is set to be the number of ones and f2 is set  to  be  the  number  of  zeros.  For   example,  if  your  ID  is  21003456,  then A1=2+1+0+0=3 and A2=3+4+5+6=18. Moreover, the 16-bit binary expression of the last 4 digits of ID, i.e., 3456, is 00001101  10000000. Thus, f1=4 Hz and f2=12 Hz. Therefore, based on this ID, the input signal is x(t)=3cos(2π*4*t)+18sin(2π*12*t), 0≤t≤1. Please generate the input signal based on your own ID and plot this input signal over the time interval 0≤t≤1 in Figure 1. Show Figure 1 in your report.

Step 2: Sampling. Set fN=2max(f1,f2) Hz. Please sample your input signal x(t) over the time interval 0≤t≤1 at two sampling frequencies: 2fN Hz and fN/2 Hz. Here, you can assume that the ideal sampling technique is employed. For each of the above two sampling rates, please plot all the samples obtained in the time interval 0≤t≤1 in Figure 2. Show Figure 2 in your reports.

Step 3: Quantization. Under each of the above two sampling rates, you need to perform uniform quantization using 4 quantization bits for each sample. For example, when the sampling rate is 2fN Hz, you get 2fN  samples from the input signal in the time interval 0≤t≤1. Please use 4 quantization bits to represent each of these samples. Note that the maximum and minimum values of x(t) areA1+A2  and –A1-A2, respectively. As a result, if you have 4 quantization bits, you can divide the regime [–A1-A2, A1+A2] into 24  sub-regimes. Suppose that natural binary coding is used for assigning quantization bits. Please refer to the lecture slides to see how to assign the quantization bits to each sub-regime. Similarly, when the sampling rate is fN/2 Hz, you get fN/2 samples from the input signal in the time interval 0≤t≤1, and you use 4 bits to quantize each sample.

Now, you have two cases: 1. sampling rate is 2fN Hz, number of quantization bits is 4; 2.  sampling  rate is  fN/2 Hz,  number  of  quantization  bits  is  4.  Please  show  your quantization codebook (how to assign 4 bits each sub-regime) in Figure 3 of your report. Then, for each of the above two cases, please show the quantization bits for all the samples in your report (suppose that natural binary coding is used).

Step 4: De-quantization. Suppose that the receiver can receive all the quantization bits without  error.  Then,  you  need  to  perform  de-quantization  based  on  the  received quantization bits using the same quantization codebook as the transmitter. For both sampling rates 2fN Hz and fN/2 Hz, please show in the report the recovered samples over the time interval 0≤t≤1 in Figure 4.

Step 5: De-sampling. Based on the recovered samples, the receiver can perform de- sampling to get the estimation of x(t) over the time interval 0≤t≤1. Please refer to the lecture slides for how to do the de-sampling. Please show your estimated signals over the time interval 0≤t≤1 at the two sampling rates in Figure 5.

Important Notes: Due to time limitation, this workshop can be finished in two parts. The first part is about Steps 1 – 4, and the second part is about Step 5. The students have to finish the first part in the lab during the 3-hour period in Workshop 1. Then, the students can copy their codes and finish the second part after Workshop 1. The second part has to be submitted to Blackboard before 11:59pm, February 24, 2025.

Submission Checklist 1 (Step 1 – Step 4; show to the TAs at the lab):

1.   Your Matlab or Pathon code

2.   Your name and student ID

3.   Figures 1 to 4 as required in Steps 1 - 4. Note that in Figures 2 and 4, each figure should show two sampling rates.

4.   The quantization bits for all the samples in Step 3, under two sampling rates.