DSP_FOEHU - Lec 07 - Digital Filters

‫ر‬َ‫ـد‬ْ‫ق‬‫ِـ‬‫ن‬،،،‫لما‬‫اننا‬ ‫نصدق‬ْْ‫ق‬ِ‫ن‬‫ر‬َ‫د‬
LECTURE (07)
Digital Filters
Assist. Prof. Amr E. Mohamed

Agenda
 Introduction
 Linear Time-Invariant Filters
 Recursive and non-Recursive Filters
 Filtering, Convolution and Correlation Operations
 Filter Structures
 Design of FIR Filters
 Design of Filterbanks
 Design of IIR Filters
 Issues in the Design and Implementation of a Digital Filter
2

Introduction
 Filters are a basic component of all signal processing and
telecommunication systems.
 Filters are widely employed in signal processing and communication
systems in applications such as channel equalization, noise
reduction, radar, audio processing, video processing, biomedical
signal processing, and analysis of economic and financial data.
 The primary functions of filters are one of the followings:
 To confine a signal into a prescribed frequency band as in low-pass,
high-pass, and band-pass filters.
 To decompose a signal into two or more sub-bands as in filter-banks,
graphic equalizers, sub-band coders, frequency multiplexers.
 To modify the frequency spectrum of a signal as in telephone channel
equalization and audio graphic equalizers.
 To model the input-output relationship of a system such as
telecommunication channels, human vocal tract, and music synthesizers.
3

Linear Time-Invariant Digital Filters
 Linear time-invariant (LTI) filters are a class of filters whose output is a
linear combination of the input signal samples and whose
coefficients do not vary with time.
 The linear property ≡ the principle of superposition.
 The time-invariant implies that the filter coefficients and hence its
frequency response is fixed and does not vary with time.
 In the time domain the input-output relationship of a discrete-time
linear filter is given by the following linear difference equation:
 

M
k
k
N
k
k knxbknyany
01
)()()(
4

Linear Time-Invariant Digital Filters (Cont.)
 Where {𝑎 𝑘 , 𝑏 𝑘} are the filter coefficients, and the output 𝑦(𝑛) is a linear
combination of the previous 𝑁 output samples [𝑦(𝑛 − 1), … , 𝑦(𝑛 − 𝑁)], the present
input sample 𝑥(𝑛) and the previous 𝑀 input samples [𝑥(𝑛 − 1), … , 𝑥(𝑛 − 𝑀)].
 The characteristic of a filter is completely determined by its coefficients
{𝑎 𝑘 , 𝑏 𝑘}.
 For a time-invariant filter the coefficients {𝑎 𝑘 , 𝑏 𝑘} are constants calculated
to obtain a specified frequency response.
 

M
k
k
N
k
k knxbknyany
01
)()()(
5

 The filter transfer function, obtained by taking the z-transform of
the general difference equation, is given by:
𝐻 𝑧 =
𝑘=1
𝑀
𝑏 𝑘 𝑧−𝑘
1 + 𝑘=1
𝑁
𝑎 𝑘 𝑧−𝑘
 The frequency response of this filter can be obtained from by
substituting the frequency variable 𝑒 𝑗𝜔
for the 𝑧 variable, 𝑧 = 𝑒 𝑗𝜔
, as
𝐻 𝑒 𝑗𝜔 =
𝑘=1
𝑀
𝑏 𝑘 𝑒−𝑗𝜔𝑘
1 + 𝑘=1
𝑁
𝑎 𝑘 𝑒−𝑗𝜔𝑘
6

 The transfer function of the LTI filter is the ratio of two polynomials in
the variable 𝑧 and may be written in a cascade form as
𝐻 𝑧 = 𝐻1(𝑧)𝐻2(𝑧)
 where 𝐻1(𝑧) is the transfer function of a feed-forward, all-zero, filter
given by
𝐻1 𝑧 =
𝑘=1
𝑀
𝑏 𝑘 𝑧−𝑘
 and 𝐻2(𝑧) is the transfer function of a feedback, all-pole, recursive
filter given by
𝐻2 𝑧 =
1
1 + 𝑘=1
𝑁
𝑎 𝑘 𝑧−𝑘
7

 Filter Order
 The order of a discrete-time filter is the highest discrete-time delay used in
the input-output equation of the filter.
 For Example, For the following Equations
 the filter order is the larger of the values of N or M.
 For continuous-time filters the filter order is the order of the highest
differential term used in the input-output equation of the filter.
 

M
k
k
N
k
k knxbknyany
01
)()()(
8

 The block diagram
implementation of the
linear time-invariant
filter.
9

Non-Recursive or Finite Impulse Response (FIR) Filters
 A non-recursive filter has no feedback and its input-output relation is given by
𝑦 𝑛 =
𝑘=0
𝑀
𝑏 𝑘 𝑥(𝑛 − 𝑘)
 The output y(n) of a non-recursive filter is a function only of the input signal x(m).
 The response of such a filter to an impulse consists of a finite sequence of M+1 samples,
where M is the filter order.
 Hence, the filter is known as a Finite-Duration Impulse Response (FIR) filter.
 Other names for a non-recursive filter include all-zero filter, feed-forward filter or
moving average (MA) filter a term usually used in statistical signal processing literature.
10

Recursive or Infinite Impulse Response (IIR) Filters
 A recursive filter has feedback from output to input, and in general its output is a
function of the previous output samples and the present and past input
samples as described by the following equation:
𝑦 𝑛 =
𝑘=0
𝑀
𝑏 𝑘 𝑥(𝑛 − 𝑘) −
𝑘=1
𝑁
𝑎 𝑘 𝑦(𝑛 − 𝑘)
 In theory, when a recursive filter is excited by an impulse, the output persists
forever.
 Thus a recursive filter is also known as an Infinite Duration Impulse Response (IIR)
filter.
 Other names for an IIR filter include feedback filters, pole-zero filters and auto-
regressive-moving-average (ARMA) filter a term usually used in statistical signal
processing literature.
 It has a number of poles corresponding to the roots of the denominator polynomial and
it may also have a number of zeros corresponding to the roots of the numerator
polynomial.
11

FIR vs IIR
 The main difference between IIR filters and FIR filters is that
 An IIR filter is more compact in that it can usually achieve a prescribed
frequency response with a smaller number of coefficients than an FIR filter.
A smaller number of filter coefficients imply less storage requirements and
faster calculation and a higher throughput. Therefore, generally IIR filters
are more efficient in memory and computational requirements than FIR
filters.
 However, it must be noted that an FIR filter is always stable, whereas an IIR
filter can become unstable (for example if the poles of the IIR filter are
outside the unit circle) and care must be taken in design of IIR filters to
ensure stability.
 Also, All IIR filter have Non-Linear Phase, while the FIR can be designed to
be Linear Phase filter or Non-Linear Phase.
12

Filter Structures: Direct, Cascade
and Parallel Forms
13

 Discussed: How to design digital filters
 How do you implement them in practice?
 In a double-precision floating-point world – no problem
 No constraints simple architecture is possible
14

Introduction
||||,
1
)( 1
1
10
az
az
zbb
zH 


 

)1()()( 10  nuabnuabnh nn
)1()()1()( 10  nxbnxbnayny
)1()()1()( 10  nxbnxbnayny Computable
Noncomputable
15
 Discussed: How to design digital filters
 How do you implement them in practice?
 In a double-precision floating-point world – no problem
 No constraints: simple architecture is possible
 Example:

Basic Operations
 Addition
 Multiplication
 Delay
+
x1(n)
x2(n)
x1(n) + x2(n)
x(n)
a ax(n)
x(n)
x(n1)z1
16

Example
x(n) +
+
b
a1
z1
z1
a2
y(n)
y(n1)
y(n2)
17
)()2()1()( 21 nbxnyanyany  Constraints:
 speed constraints
 power constraints
 Reduce total number of operations
 In particular, multiplications might
take longer than additions & takes
more power reduce their number
 Fixed-point arithmetic takes
 much less area cheaper, faster
 The area of a fixed-point parallel
multiplier is proportional to the
product of the coefficient and data
wordlengths
 reduce data or coefficient wordlengths

Higher-Order Difference Equations
 

M
k
k
N
k
k knxbknyany
01
)()()(







 N
k
k
k
M
k
k
k
za
zb
zH
1
1
1
)(
 

M
k
k
N
k
k knxbknyany
01
)()()(
18

Block Diagram Representation (Direct Form I)
 

M
k
k
N
k
k knxbknyany
01
)()()(
+
z1
z1
+
z1
+
b0
b1
bM1
bM
x(n)
x(n1)
x(n2)
x(nM)
+
z1
z1
+
z1
+
a1
aN1
aN
y(n)
y(n1)
y(n2)
y(nM)
v(n)
19

+
z1
z1
+
z1
+
b0
b1
bM1
bM
x(n)
x(n1)
x(n2)
x(nM)
+
z1
z1
+
z1
+
a1
aN1
aN
y(n)
y(n1)
y(n2)
y(nM)
v(n)



M
k
k
k zbzH
0
1 )(



 N
k
k
k za
zH
1
2
1
1
)(
Implementing
zeros
Implementing
poles
20

+
z1
z1
+
z1
+
b0
b1
bM1
bM
x(n)
x(n1)
x(n2)
x(nM)
+
z1
z1
+
z1
+
a1
aN1
aN
y(n)
y(n1)
y(n2)
y(nM)
v(n)



M
k
k
k zbzH
0
1 )(



 N
k
k
k za
zH
1
2
1
1
)(

























N
k
k
k
M
k
k
k
za
zbzHzHzH
1
0
21
1
1
)()()(
21

+
z1
z1
+
z1
+
b0
b1
bM1
bM
x(n)
x(n1)
x(n2)
x(nM)
+
z1
z1
+
z1
+
a1
aN1
aN
y(n)
y(n1)
y(n2)
y(nM)
v(n)
How many Adders?
How many multipliers?
How many delays?
22

 The direct-form structure provides a convenient method for the
implementation of FIR filters. However, for IIR filters the direct-
form implementation is not normally used due to problems with
the design and operational stability of direct-form IIR filters.
23

Block Diagram Representation (Direct Form II)
+
z1
z1
+
z1
+
b0
b1
bN1
bN
x(n)
+
z1
z1
+
z1
+
a1
aN1
aN
y(n)
w(n1)
w(n2)
w(nN)
w(n)
Assume
M = N
24

+
z1
z1
+
z1
+
b0
b1
bN1
bN
x(n)
+
z1
z1
+
z1
+
a1
aN1
aN
y(n)
w(n1)
w(n2)
w(nN)
w(n)
Assume
M = N
)()()(
1
nxknwanw
N
k
k  
)()()(
0
nwknxbny
M
k
k  



M
k
k
k zbzH
0
1 )(


 N
k
k
k za
zH
1
2
1
1
)(
25

+
z1
z1
+
z1
+
b0
b1
bN1
bN
x(n)
+
z1
z1
+
z1
+
a1
aN1
aN
y(n)
w(n1)
w(n2)
w(nN)
w(n)
Assume
M = N



M
k
k
k zbzH
0
1 )(


 N
k
k
k za
zH
1
2
1
1
)( 


















 
 



M
k
k
kN
k
k
k
zb
za
zHzHzH
0
1
12
1
1
)()()(
Implementing
poles
Implementing
zeros
26

+
z1
z1
+
z1
+
b0
b1
bN1
bN
x(n)
+
z1
z1
+
z1
+
a1
aN1
aN
y(n)
w(n1)
w(n2)
w(nN)
w(n)
Assume
M = N
How many Adders?
How many delays?
27

Block Diagram Representation (Canonic Direct Form)
+
+
+
b0
b1
bN1
bN
x(n)
+
z1
z1
+
z1
+
a1
aN1
aN
y(n)
Assume
M = N
How many Adders?
How many delays? Max (M,N)
28

Parallel Filter Structure
 An alternative to the cascade implementation described in the previous
section is to express the filter transfer function H(z), using the partial
fraction method, in a parallel form as parallel sum of a number of
second order and first order terms as
 where in general the filter is assumed to have 𝑁1 complex conjugate
poles, 𝑁1 real zeros, and 𝑁2 real poles and 𝐾 is a constant.
29
 









21
1
1
1
2
2
1
1
1
10
11
)(
N
k k
k
N
k kk
kk
za
e
zaza
zee
KzH







 N
k
k
k
M
k
k
k
za
zb
zH
1
1
1
)(




21
1
)()(
NN
k
k zHKzH

Parallel Filter Structure (Cont.)
30

Example 3: Parallel Structure of IIR Filter
 Consider the following transfer function
 Partial fraction expansion leads to
 parallel structure with 3 branches
31

Example 4: Parallel Structure of IIR Filter
 The denominator admits 3 roots: 0.0655 ± 0.5755j and 0.0492
 Recombine the two conjugate roots and by partial fraction expansion
32

General Procedure to Obtain a Parallel Structure
 Decompose 𝐻(𝑧) using a partial fraction expansion
 Combine any pair of complex conjugate poles to obtain real-valued
elements 𝐻𝑖(𝑧)
 Optional: Combine such elements further if beneficial
 Remark: Typically we limit ourselves to second-order sections
33

Main Characteristics of the Parallel Structure
 Simple to implement
 Sometimes has an advantage over the cascade realization in terms of
internally generated quantization noise, not much though (no
amplification of errors over various stages).
 Errors of coefficient quantization of 𝐻𝑖(𝑧) affects zeros of 𝐻(𝑧) longer
coefficient wordlengths required to ensure stability.
 Zeros on the unit circle in the overall transfer function are not
preserved no saving of multipliers can be obtained for filters having
such zeros.
34

Cascade Filter Structure
 The cascade implementation of a filter is obtained by expressing
the filter transfer function H(z) in a factorized form as
 where G is the filter gain and the poles (𝑝 𝑘 𝑠) and zeros (𝑧 𝑘 𝑠) are either
complex conjugate pairs or real-valued.
35







 N
k
k
k
M
k
k
k
za
zb
zH
1
1
1
)(










 21
21
1
1*11
1
1
1*11
1
)1)(1()1(
)1)(1()1(
)( N
k
kk
N
k
k
M
k
kk
M
k
k
zpzpzp
zzzzzz
GzH

Cascade Filter Structure (Cont.)
 Each bracketed term in the previous Equation is the z-transfer function
of a first or second order IIR filter:
𝐻1 𝑧 = 𝐺
𝑘=1
𝐾
𝐻 𝑘 𝑧
36
2nd Order
System
2nd Order
System
1st Order
System
1st Order
System

Cascade Filter Structure (Cont.)
 Realization of an IIR cascade structure from second order sections.
37

Example 1: Cascade Structure
 The following transfer function was considered
 Decompose as
 cascade structure with 2 sections
38

Example 2: Cascade Structure
 The denominator admits 3 roots: 0.0655 ± 0.5755j and 0.0492
 Recombine the two conjugate roots and decompose
 cascade structure with 2 sections
39

General Procedure to Obtain Cascade Structure
 Compute the poles and zeros of 𝐻(𝑧)
 Combine any pair of complex conjugate poles to obtain real-valued
elements 𝐻𝑖(𝑧)
 Optional: Combine such elements further if beneficial
 Remark: Typically we limit ourselves to second-order sections
40

Sensitivity to coefficient quantisation
 The filter coefficients are quantized
 errors in coefficient value cause errors in pole and zero positions and,
hence, filter response
 Example: Consider a filter with four poles at z = - 0.9 (close to unit
circle but stable)
41

Example: Sensitivity to coefficient quantisation
 Direct form filter would have the following denominator polynomial in
its transfer function:
 Cascade of four first-order sections:
 Assume an error of -0.06
 Direct: error -0.06 to the third coefficient, i.e. 4.86 → 4.8 roots of the
resulting polynomial are -1.5077, -0.7775 ± 0.4533j, -0.5372 Filter unstable!
 Cascade: error -0.06 to 0.9, i.e. 0.9 → 0.84 Not big deal!
 Cascade has much lower sensitivity to coefficient quantization smaller
change, only one root affected
42

Useful Tips
 Each complex root, with its inevitable conjugate, can be implemented
by a single second-order section
 Consider a root at 𝑟𝑒 𝑗𝜔 and its conjugate 𝑟𝑒−𝑗𝜔
 the real-coefficient second-order polynomial
 If interested in placing zeros
 If interested in placing poles
43

Filters with Zeros on the Unit Circle
 Many filters (FIR and IIR) have zeros on the unit circle
 Biquadratic section with 𝑟𝑒 𝑗𝜔 and its conjugate 𝑟𝑒−𝑗𝜔
 no multiplication required - very used in practice
 Implementing a high-order filter with many zeros on the unit circle as a
cascade of biquadratic sections requires fewer total multiplications
than a direct form implementation
44

Linear Phase FIR Filters
 It takes a finite time before the input to a filter propagates through the
filter and appears at the output. A delay of T seconds in time domain
appears as a phase change of 𝑒−𝑗2𝜋𝑇
in frequency domain. The time
delay, or phase change, caused by a filter depends on the type of the
filter (e.g. linear/nonlinear, FIR/IIR) and its impulse response.
 An FIR filter of order M, with M+1 coefficients, has a time delay
of M/2 samples.
 A theorem: “if h(n) represents the impulse response of a discrete time
system, a necessary and sufficient condition for linear phase is that
h(n) have finite duration N, and that it is symmetric about its
midpoint”.
 For a causal FIR filter whose impulse response begins at zero and ends
at N-1, h(n) satisfies
ℎ(𝑛) = ℎ(𝑁 − 1 − 𝑛) 𝑓𝑜𝑟 𝑛 = 0, 1, 2, … , 𝑁 − 1
46

Linear Phase FIR Filters (Cont.)
47
ℎ(𝑛) = ℎ(𝑁 − 1 − 𝑛) 𝑓𝑜𝑟 𝑛 = 0, 1, 2, … , 𝑁 − 1

 For the case where N is even,
 Let m= N-1-n in the second sum, gives:
48

 But from symmetry, h(N-1-m) = h(m), and the summation can be reversed to give
49

 For N an odd number, a similar derivation leads to:
50

Finite-Precision Number
Representation and Quantization
51

Finite-Precision Number Representation
 In a computer, numbers are represented as combinations of a finite
number of binary digits, or bits that take values of 0 and 1
 Bits are usually organized into bytes containing 8 bits or words (16 bits,
32 bits)
 Two forms are used to represent numbers on a digital computer:
 fixed-point
 floating-point
52

Fixed-point representation
 The magnitude of the number is expressed in powers of 2, with the
binary point separating positive and negative exponents B bits number
representation, value (B,A)
 Number range
 Example: a B = 12 bit number with A = 2 bits before the binary point is
in the range -2048/1024 to +2047/1024 inclusive
 All values are quantised to integer multiples of the LSB
53

Overflow
 If the result of any calculation in the filter exceeds its number range
overflow occurs.
 By default, a value slightly greater than the maximum representable
positive number becomes a large negative number, and vice versa.
 This is called wraparound; the resulting error is huge.
 In IIR filters it can result in very large amplitude "overflow oscillations".
54

Strategies to avoid overflow
 Two strategies exist
 scaling - to ensure that values can never (or hardly ever) overflow
 saturation arithmetic - to ensure that if overflow occurs its effects are
greatly reduced
55

Saturation arithmetic
 First, the results of all calculations are to full precision.
 For example, the addition of 2 (B,A) values results in a (B +1,A +1) value;
the multiplication of a (B,A) value by a (C,D) value results in a (B+C-1,A+D-
1) value.
 Then, the higher order bits of the true result are processed to detect
overflow.
 • If overflow occurs, the maximum possible positive
 value or minimum possible negative value is returned
56

Saturation arithmetic (Cont.)
 Instead of merely masking the true result to a (B,A), overflow is
detected and the maximum possible positive value or minimum possible
negative value is returned.
 Some DSP ICs incorporate saturation arithmetic hardware
57

Scaling
 Assume that the input to a filter is bounded by 𝑥(𝑛) < 𝐶
 Then its output is bounded by
 where ℎ(𝑘) is its impulse response
 𝑘=−∞
∞
ℎ(𝑘) is known as the 𝑙1 𝑛𝑜𝑟𝑚 of the filter impulse response,
easy to compute numerically
58

𝑙1scaling
 Bounded input bounded output
𝑥(𝑛) < 𝐶  𝑘=−∞
∞
ℎ(𝑘)
 If the maximum permissible output magnitude is D, overflow cannot
occur provided we scale the output by
𝐷
𝑘=−∞
∞
ℎ(𝑘)
 However if we reduce the magnitude of signals,
 The ratio of signal power to quantization noise power becomes smaller
 Scaling worsens the noise performance of the filter
59

Alternative scaling
 The input signal which gives rise to the largest possible output is
unlikely to occur in practice, so a less conservative scaling approach is
often used
 𝑙2 scaling
 frequency – response scaling
 For both saturation arithmetic is still needed as overflow is still possible
60

𝑙2 scaling
 Choose less conservative scaling based on the scale factor
𝐷
𝐶 𝑘=−∞
∞
ℎ(𝑘)
 which is the root mean square impulse response and this is known as 𝑙2
scaling
61

Frequency-response scaling
 Suppose frequency response of the filter is 𝐺(𝑒 𝑗𝜔
)
 a sine wave of frequency ω and peak amplitude 𝐶 at the input
 gives a sine wave of peak amplitude 𝐶 𝐺(𝑒 𝑗𝜔
) at the output
 To prevent overflow of a single sine wave use scaling factor
𝐷
𝐶 max
𝜔
𝐺(𝑒 𝑗𝜔)
62

Application of scaling to cascade and parallel realizations
 At each step, you must compute the impulse response or frequency
response from the input of the overall filter to the point of interest,
taking account of all scaling already included up to that point.
 The scaling at section inputs may implemented using simple binary
shifts (by using the next smaller power of 2), or by incorporating it into
the FIR coefficient scaling of the preceding section.
 For a parallel realization, scaling is computed independently for each
section, but all section outputs must be scaled by the same amount, so
the overall scaling of each section must be made the same.
 Finally scaling is applied to the final adder(s) which add together the
outputs of the parallel sections.
63

Roundoff (quantization) noise generation
 The output of a multiplier has more bits than its inputs (for example, a
16 by 16 two's complement multiplier outputs a 31-bit two's
complement value).
 To store the output it has to be (re)quantized (low order bits have to be
thrown away)
 An error called quantization noise or roundoff noise is added at that point
 The noise variance at the multiplier output, assuming rounding is used,
is 𝑞2 /12, where 𝑞 is the LSB size after quantization. (The same as for
quantization of analogue signals.)
64

Roundoff (quantisation) noise assumptions
 It is often assumed that the quantization noise at each multiplier output
is white (independent from sample to sample).
 It is also assumed that it is independent between multipliers, so that
the noise variances ("powers") add.
 (The assumption of whiteness is actually a very poor model if the signal
is narrowband, but it is reasonable for large amplitude wideband
signals. The assumption of independence can also be a poor model.)
65

Roundoff (quantisation) noise in FIR and IIR filters
 As a result:
 The quantization noise from the multipliers of an FIR filter therefore
adds white noise directly to the output signal
 In IIR filters, the white quantization noise from the feedback
multipliers filter is fed to the input of the filter, so the resulting noise
spectrum at the filter output is coloured. Its spectrum is proportional
to the square of the filter's frequency response magnitude.
66

Roundoff (quantisation) noise
 Roundoff noise level is affected by data wordlengths, filter response,
filter structure and (to an extent) by section ordering in cascade
structures.
 Further details are in specialist texts.
 Remark. DSP ICs, and some VLSI filters, provide an accumulator store of
longer wordlength than the data wordlength (e.g. a 32-bit accumulator
for a 16-bit DSP). The multiplier outputs are accumulated at the longer
wordlength, and then the accumulator output is only quantized once.
This approach significantly reduces roundoff noise.
67

Limit cycles
 Zero-input limit cycles are self-sustaining oscillations, caused by the rounding
of the results of computations.
 Example: consider the second-order filter
 This is a stable second order IIR filter with complex poles at 𝑗0.9.
 If rounding to the nearest LSB is used at the output of the multiplier, then when 𝑦(𝑛
− 2) = ±1; ± 2; ± 3; or ± 4𝐿𝑆𝐵, the computation 0.9𝑦(𝑛 − 2) will give the result
± 1; ± 2; ± 3; 𝑜𝑟 ± 4𝐿𝑆𝐵 respectively.
 Hence a limit cycle of the form 𝑦(𝑛) = 4; 0; −4; 0; 4; 0; −4; 0 (or the same pattern
with 3,2, or 1) may occur.
 Effectively, the reason is that the rounding non-linearity has increased the feedback
gain to 1, turning the system into an oscillator. 68

Limit cycles
 Limit cycles are troublesome in some applications, especially with short
data wordlengths, where the limit cycle may be relatively large. With
the longer wordlengths of DSP ICs, it is often possible to ignore limit
cycles.
 Solutions:
 One solution is to quantize toward 0 (truncation) instead of rounding the
multiplier output. But the extra roundoff noise due to truncation may
require the data wordlength to be increased by 1 or 2 bits.
 Another solution is to use certain forms of digital filters (such as Wave
filters) which do not support limit cycles. However these are
computationally more expensive.
69

Deadbands
 Consider a simple digital low-pass filter such as is commonly used for smoothing:
𝑦 𝑛 = 𝑦 𝑛 − 1 + 𝛼 𝑥 𝑛 − 𝑦(𝑛 − 1) = 1 − 𝛼 𝑦 𝑛 − 1 + 𝛼𝑥(𝑛)
 The transfer function is
𝐻 𝑧 =
𝛼
1 − (1 − 𝛼)𝑧−1
 This has unit gain at zero frequency (z = 1), and a pole at 1 – α.
 The time constant is approximately 1/α samples, for α<<1
 If 𝛼 𝑥 𝑛 − 𝑦(𝑛 − 1) < 0.5𝐿𝑆𝐵 then the multiplier output will round to zero, and
the filter output will therefore remain constant.
 Hence a constant output known as the deadband, arises. It can be up to (0.5/α)LSB
 If, for example, (1/α)=10000 to give a time constant of 10000 samples, then the
size of the LSB of the internal arithmetic must be 5000 times smaller than the
permissible size of the deadband. This implies 13 extra bits (since 212 = 4096).
70

Coefficient quantization
 In a previous section we showed that the cascade form is much less
sensitive to coefficient quantization than a high order direct form filter.
 (This is also true of the parallel form.)
 If the filter has zeros on the unit circle and is implemented using a
second-order section, the cascade realization has the advantage that
these zeros stay on the unit circle (because a coefficient b2 = 1 is
unaffected by quantization), although their frequencies may be altered.
71

 A traditional way to study the relative merits of different filter
structures was to analyze the sensitivity of the frequency response
magnitude to random (often Gaussian) perturbations of the coefficients,
and to use this as a measure of the likely sensitivity of a given structure
to coefficient quantization.
 Various structures, including Lattice and Wave filters, give even lower
sensitivity to coefficient quantization than the cascade realization.
 However, they generally require a substantially increased number of
multipliers. For a specific filter design, you should compute the actual
filter response with quantized coefficients, and then modify it if
necessary.
72

 In dedicated hardware, such as custom ICs, where there are significant
benefits from reducing coefficient wordlengths, discrete optimization
can be used to search for the finite-wordlength filter with the closest
response to a given specification.
 Some discrete optimization algorithms, including Genetic Algorithms
and Simulated Annealing, are available in software libraries
73

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