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5 Methods for Discrete Processing and Analysis of Biosignals
tion 5.113 for even values of N follows:
g(n) =
G(0) + G ( N
2 )
N
+
N
2 −1
∑
m=1
G(m)ej2π(m−N
2 )n
N
⋅ej2π N
2
n
N
N
+
+
N−1
∑
m= N
2 +1
G(m)ej2π(m−N
2 )n
N
⋅ej2π N
2
n
N
N
= 2
N [G(0)] +
N
2 −1
∑
m=1
G(m)ej2π(m−N
2 )n
N
⋅ejπn
N
+
1
∑
m= N
2 −1
G(N −m)ej2π( N
2 −m)n
N
ejπn
N
= 2
N ℜ[G(0)] +
N
2 −1
∑
m=1
{[G(m)ej2π(m−N
2 )n
N
+ G∗(m)e−j2π(m−N
2 )n
N
] (−1)n
N
}
= 2
N ℜ[G(0)] +
N
2 −1
∑
m=1
{[|G(m)|ejφej2π(m−N
2 )n
N
+ |G(m)|e−jφe−j2π(m−N
2 )n
N
] (−1)n
N
}
= 2
N {ℜ[G(0)] +
N
2 −1
∑
m=1
(−1)n|G(m)| cos [φ + 2π (m −N
2 ) n/N]}
(5.114)
and for odd values of N with analogous calculation
g(n) = 1
N {G(0) + 2
N−1
2
∑
m=1
(−1)n|G(m)| cos [φ + 2π (m −N −1
2
) n/N]} .
(5.115)
Explanatory Example
A ideal digital low-pass filter with a constant magnitude frequency response from 0
to its cut-off frequency of fg = 200 Hz and a sampling frequency of fa = 1 kHz is to
be implemented by a 15th order FIR filter using the frequency sampling method. The
periodic frequency response can be divided into N −1 = 15 −1 = 14 intervals and
sampled with 15 values. With symmetrical filter coefficients and odd filter order, this
filter has a group delay of t0 = Ta N−1
2
= 14⋅1 ms
2
= 7 ms.
The values to be specified for the magnitude of the transfer function |G(m)| with
m = 0 to N−1
2
+ 1 = 15−1
2
+ 1 = 8 are then to be specified at the frequency spacing of
fa/N = 1 kHz
15
= 66.67 Hz, viz. i.e. at frequency values f = 0 Hz, 66.67 Hz, 133.33 Hz,
200 Hz, 266.67 Hz, 333.33 Hz, 400 Hz and 466.67 Hz. After that, starting from half
the sampling frequency, the frequency response repeats mirror-symmetrically up to
the sampling frequency as shown above. If the maximum value is to be = 1, we can
specify, for example, the following values:
|G(0)| = |G(1)| = |G(2)| = |G(3)| = 1 ,
|G(4)| = |G(5)| = |G(6)| = |G(7)| = 0 .