Signal Processing - May 2016 - 116

Output of the
Notch Filter

I /P

C (z )
α
-
Output of the
Bandpass Filter

1.2

1

1

0.8

Magnitude Response

Magnitude Response

(a)

0.8
0.6
0.4

Cn (z ) with ρ = 0.9
Hn (z ) with α = 5
Hn (z ) with α = 20

0.2
0

0

0.2
0.4
0.6
0.8
Normalized Frequency ω /π
(b)

Hb (z ) with α = 5
Hb (z ) with α = 10
Hb (z ) with α = 20

0.6
0.4
0.2
0

1

0

0.2
0.4
0.6
0.8
Normalized Frequency ω /π
(c)

1

FIGure 2. (a) The general feedback structure. (b) The magnitude responses of the original notch filter C n (z) and the improved notch filters H n (z ) with

0.98

2

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1

0.96
Imaginary Part

Imaginary Part

a = 5 and 20, when t = 0.9 and ~ * = 0.4r . (c) The magnitude responses of the extremely narrow bandpass filter H b (z ) with a = 5, 10, and 20, when
t = 0.9 and ~ * = 0.4r .

-1

-0.5

0
Real Part
(a)

2
0.5

0.94
0.92

2
Original Poles
Imporved Poles

0.9
0.88
0.86

1

0.22 0.24 0.26 0.28 0.3 0.32 0.34
Real Part
(b)

FIGure 3. The pole-zero plot. (a) Poles and zeros of the original C n (z ) with t = 0.9 and H n (z ) with
a = 20 . (b) Zoomed portion.

To see how H N (z) works, the proposed feedback structure is applied to a
conventional notch filter example.
Consider the notch filter C n (z) given in
(1) with t = 0.9 and then substitute it
into C (z) in (2) and (3). The proposed
notch filter and bandpass filter are
given as
116

H n (z) =

(1 + a) C n (z)
1 + aC n (z)

(4)

and
H b (z) = 1 - H n (z) =

1 - C n (z)
. (5)
1 + aC n (z)

IEEE Signal Processing Magazine

|

May 2016

|

The frequency responses of several
notch filters H n (z) and bandpass filters
H b (z) with various a values are shown
in Figure 2(b) and 2(c), respectively. For
ease of comparison, we have included
the original simple notch filter C n (z)
with t = 0.9 in Figure 2(b).
From Figure 2(b) and (c), it is clear
that the passband of the notch filter is
nearly ideal, which implies that the
stopband of the bandpass filter is
almost zero. Moreover, the BW of
both is extremely narrow. That is, the
bandpass filter will tend to pass an
extremely narrow band signal with
frequencies in the neighborhood of
~ * while the notch filter stops the
signal with frequencies in the neighborhood of ~ *. Compared to [7],
which introduced that a narrow bandpass filter could be obtained by containing a notch filter in a feedback
path, our proposed bandpass filter not
only has an extremely narrow BW but
also preserves the near unity gain as
a enlarges.



Table of Contents for the Digital Edition of Signal Processing - May 2016

Signal Processing - May 2016 - Cover1
Signal Processing - May 2016 - Cover2
Signal Processing - May 2016 - 1
Signal Processing - May 2016 - 2
Signal Processing - May 2016 - 3
Signal Processing - May 2016 - 4
Signal Processing - May 2016 - 5
Signal Processing - May 2016 - 6
Signal Processing - May 2016 - 7
Signal Processing - May 2016 - 8
Signal Processing - May 2016 - 9
Signal Processing - May 2016 - 10
Signal Processing - May 2016 - 11
Signal Processing - May 2016 - 12
Signal Processing - May 2016 - 13
Signal Processing - May 2016 - 14
Signal Processing - May 2016 - 15
Signal Processing - May 2016 - 16
Signal Processing - May 2016 - 17
Signal Processing - May 2016 - 18
Signal Processing - May 2016 - 19
Signal Processing - May 2016 - 20
Signal Processing - May 2016 - 21
Signal Processing - May 2016 - 22
Signal Processing - May 2016 - 23
Signal Processing - May 2016 - 24
Signal Processing - May 2016 - 25
Signal Processing - May 2016 - 26
Signal Processing - May 2016 - 27
Signal Processing - May 2016 - 28
Signal Processing - May 2016 - 29
Signal Processing - May 2016 - 30
Signal Processing - May 2016 - 31
Signal Processing - May 2016 - 32
Signal Processing - May 2016 - 33
Signal Processing - May 2016 - 34
Signal Processing - May 2016 - 35
Signal Processing - May 2016 - 36
Signal Processing - May 2016 - 37
Signal Processing - May 2016 - 38
Signal Processing - May 2016 - 39
Signal Processing - May 2016 - 40
Signal Processing - May 2016 - 41
Signal Processing - May 2016 - 42
Signal Processing - May 2016 - 43
Signal Processing - May 2016 - 44
Signal Processing - May 2016 - 45
Signal Processing - May 2016 - 46
Signal Processing - May 2016 - 47
Signal Processing - May 2016 - 48
Signal Processing - May 2016 - 49
Signal Processing - May 2016 - 50
Signal Processing - May 2016 - 51
Signal Processing - May 2016 - 52
Signal Processing - May 2016 - 53
Signal Processing - May 2016 - 54
Signal Processing - May 2016 - 55
Signal Processing - May 2016 - 56
Signal Processing - May 2016 - 57
Signal Processing - May 2016 - 58
Signal Processing - May 2016 - 59
Signal Processing - May 2016 - 60
Signal Processing - May 2016 - 61
Signal Processing - May 2016 - 62
Signal Processing - May 2016 - 63
Signal Processing - May 2016 - 64
Signal Processing - May 2016 - 65
Signal Processing - May 2016 - 66
Signal Processing - May 2016 - 67
Signal Processing - May 2016 - 68
Signal Processing - May 2016 - 69
Signal Processing - May 2016 - 70
Signal Processing - May 2016 - 71
Signal Processing - May 2016 - 72
Signal Processing - May 2016 - 73
Signal Processing - May 2016 - 74
Signal Processing - May 2016 - 75
Signal Processing - May 2016 - 76
Signal Processing - May 2016 - 77
Signal Processing - May 2016 - 78
Signal Processing - May 2016 - 79
Signal Processing - May 2016 - 80
Signal Processing - May 2016 - 81
Signal Processing - May 2016 - 82
Signal Processing - May 2016 - 83
Signal Processing - May 2016 - 84
Signal Processing - May 2016 - 85
Signal Processing - May 2016 - 86
Signal Processing - May 2016 - 87
Signal Processing - May 2016 - 88
Signal Processing - May 2016 - 89
Signal Processing - May 2016 - 90
Signal Processing - May 2016 - 91
Signal Processing - May 2016 - 92
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Signal Processing - May 2016 - 94
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Signal Processing - May 2016 - 96
Signal Processing - May 2016 - 97
Signal Processing - May 2016 - 98
Signal Processing - May 2016 - 99
Signal Processing - May 2016 - 100
Signal Processing - May 2016 - 101
Signal Processing - May 2016 - 102
Signal Processing - May 2016 - 103
Signal Processing - May 2016 - 104
Signal Processing - May 2016 - 105
Signal Processing - May 2016 - 106
Signal Processing - May 2016 - 107
Signal Processing - May 2016 - 108
Signal Processing - May 2016 - 109
Signal Processing - May 2016 - 110
Signal Processing - May 2016 - 111
Signal Processing - May 2016 - 112
Signal Processing - May 2016 - 113
Signal Processing - May 2016 - 114
Signal Processing - May 2016 - 115
Signal Processing - May 2016 - 116
Signal Processing - May 2016 - 117
Signal Processing - May 2016 - 118
Signal Processing - May 2016 - 119
Signal Processing - May 2016 - 120
Signal Processing - May 2016 - Cover3
Signal Processing - May 2016 - Cover4
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