FIR Filters

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FIR Filters
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2.1 Introduction
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Figure 2.2 Unit impulse function (a) and
example of a causal and stable FIR impulse
response
(n)
a) b)
n

0 1 2 3 4 5
h(n)
0

n
1
2


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2.2 Transversal Structures
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2.2.1 Direct Form for the GeneralImpulse Response
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Figure 2.3 First direct form of a transversal
filter
u(n) u(n) u(n-1) u(n-2) u(n-3)
u(n-N1)
z-1
z-1
z-1
z-1
z-1
hN-1
h3
h2
hN-2
h1
h0
y(n)
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2.2.2 Direct Form with Symmetrical Impulse
Response
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2.2.3 Transposed Direct Form with General Impulse
Response
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Figure 2.8 Transposed direct form
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2.2.4 Transposed Direct Form with Symmetrical
Impulse Response
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2.3 Lattice Structures
Lattice structures were first used in
autoregressive signal models. When dealing with
filter banks, their most frequent application is
the realisation of robust systems with perfect
reconstruction.
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2.3.1 Standard Lattice structures
Xi-1(z)
Xi(z)
Yi(z)
Yi-1(z)
Figure 2.10 Basic lattice stage
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Figure 2.11 Lattice cascade, with one input and
two outputs (analysis structure)
X2(z)
XN-2(z)
XN-1(z)
X1(z)
U(z)
section N-1
section N-2
section 2
section 1
Y1(z)
YN-1(z)
Y2(z)
YN-1(z)
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Let us now define two transfer functions for the
lattice structure in Fig. 2.11Hi(z) Xi(z) /
U(z), i 1,2,3.....N-1 (2.9) Gi(z)
Yi(z) / U(z), i 1,2,3.....N-1
(2.10)With these equations, we can use
induction to derive the following recursive
equations
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Example 2.1
Consider the lattice structure in fig 2.12 What
is the transfer function H(z)X(z)/U(z)? The
structure in fig. 2.12 can be dealt with from
left to right. With the lattice coefficient
q10.9, q20.8 and q3 0.7, and with Ho(z)
Go(z)1, we obtain
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Figure 2.12 Example of a three-stage lattice
FIR filter
U(z)
X(z)
0.9 0.8
0.7
0.9 0.8
0.7
z-1 z-1
z-1
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The required transfer function is given by H(z)
H3(z).
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Synthesis of the lattice H(z) known, qi ?
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we will have
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Example 2.2
Derive a lattice structure which has the transfer
function
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Next, it is necessary to formulate G3(z) from
H3(z) H(z),as in 2.12
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2.3.2 QMF LatticesQMF lattices are usually used
for implementing paraunitary quadrarature mirror
filter banks Vai 88. They have two special
characteristics

• In each stage of the lattice, one coefficient is
  positive, and the other negative, but both with
  same magnitude, and
• all coefficients with even-valued indices are
  zero.

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These special properites limit the type of
transfer functions that can be realised. With the
lattice structure in Fig. 2.13, only so-called
power complementary conjugate quadrature filters
can be implemented, see also section 6.3.2. For
this type of transfer function, we can say, using
the definitions in (2.9) and (2.10)
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Figure 2.13 QMF lattice structure
U(z) X1(z)
X3(z) X5(z) XN-1(z)
-q1
-q5
-qN-1
-q3
q1
q3
q5
qN-1
z-2
z-2
z-2
z-1

Y1(z) Y3(z) Y5(z)
YN-1(z)
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Where N is assumed to be even, and
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Equation (2.26) can be used with (2.27) or (2.25)
to analyse QMF lattice structres. In this case,
the analysis proceeds from the input to the
output.
Synthesis procedure is done using
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Example 2.3
What is the transfer function H(z) X(z)/U(z) of
the lattice structure in the Fig2.14? The
transfer function of first stage can be obtained
directly H1(z)
1-0.4z-1
G1(z) 0.4z-1
Substitution confirms that the first stage
satisfies equations (2.24) and (2.25). The
required transfer function H(z) H3(z) can be
found using (2.26) H(z) H3(z) H1(z) - 0.7 .
z-2 G1(z) 1- 0.4 z-1-
0.28z-2 0.7 z-3
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Figure 2.14 Example of a QMF lattice structure
U(z)
X(z)
-0.4
-0.7
0.7
0.4
z-1
z-2
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2.4 Symmetry Properties and Linear Phase
2.4.1 Zero-Phase, Even Prototypes
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Substitute m by n and offset 1
(-1)k
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In what follows, we shall make the substiution m
n and describe prototype 2 using
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2.4.2 Zero-phase, Odd prototypes
Prototypes 3 and 4 are derived from the odd
impulse response
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Finally, let us consider prototype 4, which is
formed by downsampling the odd signal in
Fig.2.15b with a phase offset ?1
These four non-causal, zero-phase prototypes will
be used in the following two sections to derive
four different, causal, linear-phase FIR filters.
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2.4.3 Linear-Phase FIR Filters
The prototypes looked at so far have all had an
even or odd impulse response, and are thus
non-causal. However, we can derive casusal FIR
filter from them by shifting the impulse response
in positive direction. The size of this shift is
half the distance between the first and last
cofficients
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Prototypes a1(n) and a3(n), which have an odd
number of coefficients, are shifted by an
integral number of steps, Fig.2.20. Prototypes
a2(n) and a4(n), with an even number of
coefficients, are shifted by an integral number
plus an extra half step. These two prototypes
fit into the integer time grid, Fig. 2.21.
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For i 3,4, the extra phase offset ?/2 is caused
by the imaginary factor j. In all four cases,
the group delay can be obtained by differerenting
(2.48) with respect to the frequency ?
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2.5 Complementary Filters
Complementing can be used to derive a new
transfer function from an old one. This process
will often be used in the following chapters.

• 2.5.1 Zero phase Complementary Filters

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Figure 2.22 Zero-phase complementary filter
struture
1
U(z)
Y(z)
_
A(z)
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Analysis of the structure in Fig. 2.22
givesY(z) U(z) A(z)U(z)
1- A(z) U(z) (2.53)
Ac(z) U(z)The complementary transfer function
Ac(z) 1-A(z)
(2.54)is the ones complement of the transfer
function A(z).
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Example
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Substituting z ej? into (2.54), we obtain the
real complementary frequency response Ac(ej?),
that is shown in Fig. 2.24b. From (2.54)
Ac(ej ?) A(ej ? ) 1 (2.55)
At any frequency ?, the sum of the frequency
response Ac(ej ?) and A(ej ? ) is 1. They are
complements of each other. Comparing Figs. 2.33
and 2.24 shows that complementing a low-pass
filter gives a high-pass filter. The inverse
procedure is also possible.
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In the time-domain, using ac(n)
Ac(z), ?(n) 1 and a(n)
A(z) with (2.54) we obtain

• ac(n) ?(n)-a(n)
  (2.56)
• When taking the complement, the impulse response
  is negated, and a one is added at n0, as in Fig.
  2.24a. This whole process assume that the
  impluse response have an odd number N of
  coefficients.

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2.5.2 Causal Complementary Filters

• Taking the complement of the systems with a real
  frequency response, which has been explained
  above, can now be easily expanded to causal,
  linear-phase systems. For this, all signals must
  be shifted by half the lenght of the impulse
  response, as shown in fig.2.25. Assuming that
  there are N cofficients,
• where N is odd, (2.56) gives

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Taking the complement, the causal impulse
response is thus negated, and a one is added at
the point n (N-1)/2, the middle of the impulse
response. The even symmetry about the mid-point
of the impulse response is preserved and the
filter keeps its linear-phase property.
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The transfer function Hc(z) of the complementary
filter can be derived from the z-transform of
(2.59). With the corresponding H(z)
h(n),
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Figure 2.26 Causal complementary filter
structure
z-(N-1)/2
U(z)
Y(z)
_
H(z)
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Fig 2.26 shows the corresponding filter stucture.
The signal paths from the input to the output
node both have the same propagation delay
(N-1)/2. In the frequency response of the
complementary filter, this delay can be taken out
as a factor. Allowing for (2.58), we have
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The amplitude frequency response of the causal
complementary filter, 1-A(e j?) , is the same as
for non-causal prototype, see (2.55). From
(2.61) it can also be seen again that the
complementary filter is a linear-phase filter
with group delay (N-1)/2.
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The high- pass/lowpass and lowpass /highpass
transformations are the most important
applications of complements, and the most widely
used. However, there are some other uses. In
principle, the complement can be used in
conjunction with any frequency response. For
example, a bandpass filter can easily be
transformed to a band-stop filter.
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2.6 Half-Band Filters

• Half-Band pass filters can be used to model
  crosstalk in filter banks
• Extension M-th band filters

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2.6.1 Zero-Phase Half-Band Low-Pass Filters

• The following discussion starts by considering
  a non-causal FIR prototype with an even impulse
  response
• a(n) A(z).
• Splitting A(z) into two polyphase components
  gives

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The ripple values dp and ds in the pass-band and
stop-band are equal dp ds
(2.68)
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• In cases where the conditions in (2.67) and
  (2.68) are acceptable, the reduction in the
  number of filter operations can be exploited.
  Instead of N multiplications, only (N3)/ 2 need
  to be performed.

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2.6.2 Causal Half-Band Low-Pass
Filters
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Since, for the number of coefficients N of a
half-band filter,N 4i 1 , i integer
positive, (2.70)(N-1)/2 is always
odd. The polyphase components are thus swapped
with each other during the transition to a causal
half-band filter. Fig. 2.30 shows a structure to
realise an FIR filter.
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Figure 2.30 Realising a half-band filter main
structre (a) and signal flow graph (b)
u(n) y(n)
a) b)
z-1
h5
u(n)
y(n)
h0
h4
h2
h8
h6
h10
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2.6.3 Half-Band Band-Pass Filters
(HB2P-Filters)

• Crosstalk
• Characteristic of orthognal filters in banks

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Even real and odd imaginary partDefineFirst
polyphase component Type 1.
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Figure 2.31 symmetry properties of half-band
band-pass filter real part (a) and imaginary
part (b) of an HB2P type 1 filter, and real (c)
and imaginary (d) parts of HB2P type 2 Filter
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a) b) c) d)
0
0
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of the HB2P type 1 filter has a characteristic
that will be very important later on. Since the
function only contain even powers of
z, all odd-numbered coefficients of the impulse
response are zero
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Since the real part of is an even
function of ?, it follows from (2.79) that there
is odd symmetry about the half-band frequency ?
? /2 . The imaginary part of is an
odd function of ?. From (2.79), it follows that
there is even symmetry about the half-band
frequency ? ?/2. These symmetry relationships
are shown in Figs. 2.31 c and d.
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Fig. 2.31 also shows that the frequency response
of the type 1HB2P filter has a period ?, while
the type 2 HB2P filter has a period of 2
?.Finally, it can be seen from (2.73) that the
transfer function only contains odd
powers of z. The even-numbered coefficients of
the corresponding impulse response are thus zero
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2.6.4 HB2P Filters from low-pass prototypes

• This section describes how to derive HB2P filter
  from FIR low-pass proto-types. Let us first
  consider the transfer function of an arbitrary
  FIR filter with real coefficients. Its polyphase
  representation with M2 is

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The transfer function of the corresponding
high-pass can be expressed as
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Fig.2.32 shows as an example a low-pass prototype
with 33 coefficients that has been designed using
the Parks-McClellan design. Fig. 2.33 shows a
high pass filtr that has been derived from this
prototype using (2.82).
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2.6.5 M-th-Band Low-Pass Filters

• The M-th-Band Low-Pass Filters is a
  generalisation of the half-band low-pass filter,
  and is used to design M-band filter banks. A
  low-pass filter with the polyphase representation

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The sum of the M copies of these shifted filters
is 1!
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Due to (2.87), the transfer function contains no
powers that are an integer multiple of M. Thus,
for the impulse response of the M-th-band
low-pass filter