ECE310 Digital Signal Processing

Problem Set IV: Quantization

1. In this problem, we deÖne “truncation” to mean simply discarding the LSBs on the

right, without adjusting the other bits that are kept. Also, we assume rounding operations are performed Örst, followed by overáow operations. For example, Örst round up

if necessary, then test for overáow.

Consider the following twoís complement Öxed point values:

00110011:010101001

110101010101:1010101

Do not compute the numerical values. Just produce the twoís complement codes as

speciÖed. In all cases, show the code AFTER you have performed roundo§ but BEFORE you have applied the overáow operation, and then show the FINAL ANSWER.

(a) Produce the (5).(4) codes assuming: roundo§ by rounding with twoís complement

overáow; and roundo§ by truncation with saturation overáow.

(b) Produce the (4).(3) codes assuming roundo§ by rounding with twoís complement

overáow; and roundo§ by truncation with saturation overáow.

2. For each of the following twoís complement values, write the code with the fewest

number of bits that can represent the same value exactly. Do this without computing

the numerical values.

0001011:011100

11111111111101:01100

3. Now compute the numerical values for your answers to 2, left in the form of a fraction

3

3

4

, etc.

:

4. Sensitivity Properties of Parallel Allpass Realizations

When a Ölter is decomposed as the sum of two lower order Ölters, e.g., H (z) =

H1 (z) + H2 (z), it is called a parallel conÖguration; this contrasts with the cascade

conÖguration which corresponds to H (z) = H1 (z) H2 (z). It turns out that a large

class of digital Ölters (including Butterworth, Chebyshev and elliptic Ölters) can be

realized by a parallel pair of allpass Ölters. In general, the allpass Ölters will have

complex coe¢ cients, even if the overall Ölter transfer function has real coe¢ cients.

However, there are certain situations where the allpass functions can have purely real

coe¢ cients, and that is the case in the example you will explore here. It turns out

using lossless building blocks in DSP structures, as in this case, can provide signiÖcant advantages in terms of quantization e§ects. Here you will explore the sensitivity

advantages of parallel allpass realizations.