Real-time digital signal processing: implementations, ... changes in the input signal is limited by its internal clock rate, so that it may be slow to
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From (7.1.4), we have
W
ÀkN
N
e
j2p=NkN
e
j2pk
1:
9:3:2
Multiplying the right-hand side of (9.3.1) by W
ÀkN
N
, we have
Xk W
ÀkN
N
X
NÀ1
n0
xnW
kn
N
X
NÀ1
n0
xnW
ÀkNÀn
N
:
9:3:3
Define the sequence
y
k
n
X
NÀ1
m0
xmW
ÀknÀm
N
,
9:3:4
this equation can be interpreted as a convolution of the finite-duration sequence x(n),
0 n N À 1, with the sequence W
Àkn
N
un.
Consequently, y
k
n can be viewed as the output of a filter with impulse response
W
Àkn
N
un. That is, the filter with impulse response
h
k
n W
Àkn
N
un
9:3:5
due to the finite-length input x(n). Thus Equation (9.3.4) can be expressed as
y
k
n xn à W
Àkn
N
un:
9:3:6
From (9.3.3) and (9.3.4), and the fact that xn 0 for n < 0 and n ! N, we show that
Xk y
k
nj
nNÀ1
:
9:3:7
That is, X(k) is the output of filter H
k
z at time n N À 1.
Taking the z-transform of (9.3.6) at both sides, we obtain
Y
k
z Xz
1
1 À W
Àk
N
z
À1
:
9:3:8
The transfer function of the kth Goertzel filter is defined as
H
k
z
Y
k
z
Xz
1
1 À W
Àk
N
z
À1
, k 0, 1, . . . , N À 1:
9:3:9
This filter has a pole on the unit circle at the frequency !
k
2pk=N. Thus the entire
DFT can be computed by filtering the block of input data using a parallel bank of N
filters defined by (9.3.9), where each filter has a pole at the corresponding frequency of
the DFT. Since the Goertzel algorithm computes N DFT coefficients, the parameter N
412
PRACTICAL DSP APPLICATIONS IN COMMUNICATIONS
Summary :
From (7.1.4), we have W ÀkN N e j2p=NkN e j2pk 1: 9:3:2 Multiplying the right-hand side of (9.3.1) by W ÀkN N , we have Xk W ÀkN N X NÀ1 n0 xnW kn N X NÀ1 n0 xnW ÀkNÀn N : 9:3:3 Define the sequence y k n X NÀ1 m0 xmW ÀknÀm N , 9:3:4 this equation can be interpreted as a convolution of the finite-duration sequence x(n), 0 n N À 1, with the sequence W Àkn N un.
Tags :
àkn,filter,sequence,nà1,dft,934,xnw,response,defined,frequency,pole,939,936