Computer Science Assignment .

Computer Science Assignment
Student Name
Institution Affiliation

QUESTION 1
a) A signal extending from 0 to 15.0 kHz is sampled at a rate of 20% greater
than the minimum sampling rate given by the sampling theorem. Each
sample is coded as an 8-bit binary word and transmitted as a sequence of
eight binary symbols.
i. How many bits per second are generated by this process?
According to the sampling theorem, the sampling frequency should be at least
twice the highest frequency in a signal. That said, a 20% margin higher than
the highest frequency signal has been provided.
Thus, the sampling rate will be given as follows;
Sampling Rate = 120*30/100 = 36.0kHz.
Bitrate = Bits per second*Samples per second*No. of channels;
=8*36000*8
Bits per second (bit rate) = 2,304,400 bits per second
You can divide the bit rate by 1000 so that you have 2,304.4kbps
ii. Calculate the minimum bandwidth required to transmit bits at this
rate over a noise-free channel?
The minimum bandwidth is equal to the Nyquist bandwidth.
Therefore the minimum bandwidth (BW)min=W=Rb/2= (2304.4/2) =1152.2
(Cheng, Ruan, and Wu, 2005)

b. Sketch a signal constellation diagram for 16-QAM. Label the axes of
your diagram and explain what they stand for. State, with a brief explanation,
how many bits each point represents.
A rectangular QAM modulation is an equivalent of superimposing two ASK
signals on quadrature carriers. The size of each symbol is given by the formula
k=log2(M). Therefore, for a 16-QAM modulation:
k=log2(16) = 4
The next step is to generate a gray coded 16-QAM constellation using K-Maps
Using a 4 variable K-
Map (given that the symbol size as calculated above is 4), the following table is
obtained:
AB/CD 00 01 11 10
00 0000 0001 0011 0010
01 0100 0101 0111 0110
11 1100 1101 1111 1110
10 1000 1001 1011 1010
first K-Map table
Replacing AB and CD with four amplitude levels {-3, -1, +1, +3} gives the following
table:
(What you do with the table I have provided up there is just to replace those digital
digits with those amplitude levels were -3(00), -1(01), +1(11), and +3(10)

AB/CD -3 -1 +1 +3
-3 0000(0) 0001(1) 0011(3) 0010(2)
-1 0100(4) 0101 (5) 0111(7) 0110(6)
+1 1100(12) 1101(13) 1111(15) 1110(14)
+3 1000(8) 1001(9) 1011(11) 1010(10)
Assigning the equivalent decimals value for the binary digits gives the following table
Previously we had a table with only binary numbers. Then you were to replace the
binary values on the AB and CD with amplitude levels -3 to +3. Having done that,
you assign decimals equivalents to the binary digits inside the table as in the last table
I have just uploaded
The following is the 16-QAM modulator output that is then plotted as below