Scaling

1.

a) The impulse response is

b) There are values that when taken the absolute value they exceed one. See h13.m.

c) Assuming input values between [-1,1] we must find a scaling term K such that (see lecture notes page 30 / Worst-case scaling). To not to change the overall transfer function we must place a multiplier 1/K at the output.

See h9.m

d) See h13.m

e) Using a step function causes an overflow at the output of H(z) if K is higher than 1/5.

f) Scaling H(z) according to norm means finding K such that

See h13.m

Overflows occur at approximately 10% rate for uniformly distributed noise.

g) There are two error sources, one after the scaling factor and another after the multiplier in the feedback loop. The output noise variance is (with help of exercise 12)

and using 1+6 bits we get