the power spectral density at the filter output is white
within the bandwidth
B
N
and zero elsewhere, forming an
equivalent rectangular spectral density;
(b) the area under this rectangular spectral density is equal to
the area under the spectral density at the filter output.
ERG2310AI
p. I77
Equivalent Noise Bandwidth
Let
ω
0
be the midband frequency of the system
,
.
)
(
)
(
2
2
1
)
(
2
0
2
2
2
0
2
N
B
B
o
B
H
d
H
t
n
N
N
ω
η
ω
ω
η
π
π
π
=
=
∫
−
2
0
0
2
)
(
)
(
2
1
ω
ω
ω
π
H
d
H
B
N
∫
∞
=
⇒
∫
∫
∞
∞
∞
−
=
=
0
2
2
2
)
(
2
)
(
2
2
1
)
(
ω
ω
π
η
ω
ω
η
π
d
H
d
H
t
n
o
Q
ω
o
ω

Η(ω
o
)
2
2πΒ
Ν
Η(ω
)
(
ω
o
= 0
for lowpass filter)
ERG2310AI
p. I78
Equivalent Noise Bandwidth
Example: Computer the –3dB bandwidth and the equivalent noise bandwidth
of a filter with the following magnitude transfer characteristic:
4
1
1
)
(
ω
ω
+
=
H
As 
H(0)
=1, the –3dB bandwidth is found by solving
2
1
)
(
3
=
dB
H
ω
Æ
ω
3dB
=1
or
f
3dB
=1/(2
π
) = 0.159Hz
∫
∫
∞
∞
=
=
+
=
=
0
4
2
0
0
2
177
.
0
8
2
1
1
2
1
)
(
)
(
2
1
Hz
d
H
d
H
B
N
ω
ω
π
ω
ω
ω
π
Using
ERG2310AI
p. I79
SignaltoNoise Ratio (SNR)
It is a dimensionless ratio of signal power to noise power
.
)
(
/
)
(
/
2
2
t
n
t
s
N
S
=
It is quite common to express the signaltonoise ratio in
decibels
[
]
[
]
.
)
(
/
)
(
log
10
/
2
2
10
t
n
t
s
N
S
dB
=
* Both of these meansquare values assume zero mean value unless
stated otherwise.
SNR is commonly used to describe the quality of a signal.
ERG2310AI
p. I80
Noise Figure
Let the input and output signal voltages (or currents) in a given system
be
s
i
(
t
),
s
o
(
t
)
respectively and the input and output noise voltages (or
currents) be
n
i
(
t
),
n
o
(
t
).
The input signaltonoise ratio is
,
)
(
/
)
(
)
/
(
2
2
t
n
t
s
N
S
i
i
i
=
The output signaltonoise ratio is
,
)
(
/
)
(
)
/
(
2
2
t
n
t
s
N
S
o
o
o
=
The
noise figure
F
is defined to be the ratio of the input signalto
noise ratio to the output signaltonoise ratio:
.
)
/
(
)
/
(
o
i
N
S
N
S
F
=
(S/N)
o
h
(
t
)
(S/N)
i
Noise figure (expressed in decibel, i.e. 10log(
F
)) is commonly used to
describe the noise performance of a system.
For ideal system (no noise)
Î
F
=1
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 Fall '09
 Frequency, Signal Processing, lim, Autocorrelation, LTI system theory