1. With the help of the table on page 2 of the lecture notes, add the following numbers using Two’s complement, Sign and Magnitude and One’s-Complement methods: 7/8+3/8+2/8-6/8. What was your observation?

Sign & Magn.	Binary Num.
7/8	0111
3/8	0011
2/8	0010
-6/8	1110
= 6/8	= (1)1010 = -2/8 ≠ 6/8

2’s Complem.	Binary Num.
7/8	0111
+3/8	+ 0011
+2/8	+ 0010
-6/8	+ 1010
= 6/8	= (1)0110 = 6/8

1’s Complem.	Binary Num.
7/8	0111
+3/8	+0011
+2/8	+0010
-6/8	+1001
= 6/8	= (1)0101 = 5/8 ≠ 6/8

In two’s complement arithmetic the overflows don’t mess up the result, if the final result is within the desired range.

2. Generate 100000 samples of a random signal in Matlab whose mean is 0.5 and variance is 0.8. Pass it through the system h[n] = [1 2 3]. First calculate (using pen and paper) what the mean and variance of the output should be. Next find out the mean and variance of the output noise using Matlab. Is there any difference between what you calculated and what Matlab suggests?

See H12.m.

Using formulas on page 22 we get

There is a slight difference between the theoretical results and the results obtained in Matlab, depending on the length of the random vector (the longer the vector, the more accurate the results).

3. Prove: (lecture notes, page 17) .

4. Consider the system: y[n]=1.2x[n]+0.5y[n-1]

Depict the statistical model for fixed-point round-off error of this system. Express the mean and the variance of the output round-off error in terms of the mean and the variance of the input round-off error (denoted by m_e and σ_e² respectively). What is the noise gain of this system? Noise gain is defined on page 63 of the lecture notes.

,

where is solved with help of H(z) (H(z) is the transfer function after the multiplier 1.2) and because the errors go through the same system.

.

Because the transform of is , .

So

and

Noise gain is

5. How many bits are required for SNR ≥ 60 dB?

Signal-to-noise ratio:

for

If the scaling is performed such that , we get

If we use the other equation (), we get