effects of mixing, integration and differentiation of noise are studied and noise bandwidth is defined
Mixing of noise with a sinusoid
If
k
th
size 12{k rSup { size 8{ ital "th"} } } {} component of noise is mixed with a sinusoid
n
k
t
cos
2πf
o
t
=
a
k
2
cos
2π
kΔf
+
f
o
t
+
b
k
2
sin
2π
kΔf
+
f
o
t
+
a
k
2
cos
2π
kΔf
−
f
o
t
+
b
k
2
sin
2π
kΔf
+
f
o
t
alignl { stack {
size 12{n rSub { size 8{k} } left (t right )"cos"2πf rSub { size 8{o} } t= { {a rSub { size 8{k} } } over {2} } "cos"2πleft (kΔf+f rSub { size 8{o} } right )t+ { {b rSub { size 8{k} } } over {2} } "sin"2πleft (kΔf+f rSub { size 8{o} } right )t} {} #
+ { {a rSub { size 8{k} } } over {2} } "cos"2πleft (kΔf - f rSub { size 8{o} } right )t+ { {b rSub { size 8{k} } } over {2} } "sin"2πleft (kΔf+f rSub { size 8{o} } right )t {}
} } {}
Sum and difference frequency noise spectral components with 1/2 amplitude are generated and
G
n
f
+
f
o
=
G
n
f
−
f
o
=
G
n
f
4
size 12{G rSub { size 8{n} } left (f+f rSub { size 8{o} } right )=G rSub { size 8{n} } left (f - f rSub { size 8{o} } right )= { {G rSub { size 8{n} } left (f right )} over {4} } } {}
Considering power spectral components at
kΔf
size 12{kΔf} {} and
lΔf
size 12{lΔf} {} , let the mixing frequency be
f
0
=
k
+
l
Δf
size 12{f rSub { size 8{0} } = { size 8{1} } wideslash { size 8{2} } left (k+l right )Δf} {} . This will generate 2 difference frequency components at the same frequency
pΔf
=
f
0
−
kΔf
=
lΔf
−
f
0
size 12{pΔf=f rSub { size 8{0} } - kΔf=lΔf - f rSub { size 8{0} } } {}
Then difference frequency components are
n
p1
t
=
a
k
2
cos
2πpΔ
ft
−
b
k
2
sin
2πpΔ
ft
size 12{n rSub { size 8{p1} } left (t right )= { {a rSub { size 8{k} } } over {2} } "cos"2πpΔital "ft" - { {b rSub { size 8{k} } } over {2} } "sin"2πpΔital "ft"} {}
n
p2
t
=
a
l
2
cos
2πpΔ
ft
+
b
l
2
sin
2πpΔ
ft
size 12{n rSub { size 8{p2} } left (t right )= { {a rSub { size 8{l} } } over {2} } "cos"2πpΔital "ft"+ { {b rSub { size 8{l} } } over {2} } "sin"2πpΔital "ft"} {}
But as
a
k
a
l
¯
=
a
k
b
l
¯
=
b
k
a
l
¯
=
b
k
b
l
¯
=
0
size 12{ {overline {a rSub { size 8{k} } a rSub { size 8{l} } }} = {overline {a rSub { size 8{k} } b rSub { size 8{l} } }} = {overline {b rSub { size 8{k} } a rSub { size 8{l} } }} = {overline {b rSub { size 8{k} } b rSub { size 8{l} } }} =0} {}
We find
E
n
p1
t
n
p2
t
=
0
size 12{E left [n rSub { size 8{p1} } left (t right )n rSub { size 8{p2} } left (t right ) right ]=0} {}
And
E
n
p1
t
+
n
p2
t
2
=
E
n
p1
t
2
+
E
n
p2
t
2
size 12{E left lbrace left [n rSub { size 8{p1} } left (t right )+n rSub { size 8{p2} } left (t right ) right ] rSup { size 8{2} } right rbrace =E left lbrace left [n rSub { size 8{p1} } left (t right ) right ]rSup { size 8{2} } right rbrace +E left lbrace left [n rSub { size 8{p2} } left (t right ) right ] rSup { size 8{2} } right rbrace } {}
Thus superposition of power applies even after shifting due to mixing.
Minimizing noise in systems by filtering:
Assume white noise with
G
n
f
=
η
2
size 12{G rSub { size 8{n} } left (f right )= { {η} over {2} } } {}
To minimize noise entering the demodulator, a filter of bandwidth B can be placed, with B just wide enough to pass signal of interest. Output noise depends on the filter used.
Ideal LPF with white noise has
N
o
=
ηB
size 12{N rSub { size 8{o} } =ηB} {}
rectangular BPF with white noise has
N
o
=
2
η
2
f
2
−
f
1
=
η
f
2
−
f
1
size 12{N rSub { size 8{o} } =2 { {η} over {2} } left (f rSub { size 8{2} } - f rSub { size 8{1} } right )=ηleft (f rSub { size 8{2} } - f rSub { size 8{1} } right )} {}
Rc low pass filter:
The filter transfer function is
H
f
=
1
1
+
j
f
f
c
size 12{H left (f right )= { {1} over {1+j { {f} over {f rSub { size 8{c} } } } } } } {}
Using
G
no
f
=
∣
H
f
∣
2
G
ni
f
size 12{G rSub { size 8{ ital "no"} } left (f right )= lline H left (f right ) rline rSup { size 8{2} } G rSub { size 8{ ital "ni"} } left (f right )} {}
we have
G
no
f
=
η
2
1
1
+
f
f
c
2
size 12{G rSub { size 8{ ital "no"} } left (f right )= { {η} over {2} } { {1} over {1+ left ( { {f} over {f rSub { size 8{c} } } } right ) rSup { size 8{2} } } } } {}
and noise power at filter o/p is
N
o
=
∫
−
∞
∞
G
n
f
df
=
η
2
∫
−
∞
∞
df
1
+
f
f
c
2
size 12{N rSub { size 8{o} } = Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } {G rSub { size 8{n} } left (f right ) ital "df"= { {η} over {2} } Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { { { ital "df"} over {1+ left ( { {f} over {f rSub { size 8{c} } } } right ) rSup { size 8{2} } } } } } } {}
using
x
=
f
f
c
size 12{x= { {f} over {f rSub { size 8{c} } } } } {} and noting that
∫
−
∞
∞
dx
1
+
x
2
=
π
size 12{ Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { { { ital "dx"} over {1+x rSup { size 8{2} } } } } =π} {} , we have
N
o
=
π
2
ηf
c
size 12{N rSub { size 8{o} } = { {π} over {2} }ηf rSub { size 8{c} } } {}
Differentiating filter:
The transfer function is
H
f
=
j2
πτ
f
size 12{H left (f right )=j2 ital "πτ"f} {}
white noise creates output psd
G
no
f
=
4π
2
τ
2
f
2
η
2
size 12{G rSub { size 8{ ital "no"} } left (f right )=4πrSup { size 8{2} }τrSup { size 8{2} } f rSup { size 8{2} } { {η} over {2} } } {}
and following this by a rectangular lpf of bandwidth B, noise at o/p is
N
o
=
∫
−
B
B
G
no
f
df
=
4π
2
3
ητ
2
B
3
size 12{N rSub { size 8{o} } = Int cSub { size 8{ - B} } cSup { size 8{B} } {G rSub { size 8{ ital "no"} } left (f right ) ital "df"= { {4πrSup { size 8{2} } } over {3} } ital "ητ" rSup { size 8{2} } B rSup { size 8{3} } } } {}
Integrating filter:
An integrator integrating over an interval T has transfer function
H
f
=
1
j
ωτ
−
e
−
jωT
j
ωτ
size 12{H left (f right )= { {1} over {j ital "ωτ"} } - { {e rSup { size 8{ - jωT} } } over {j ital "ωτ"} } } {}
and thus
∣
H
f
∣
2
=
T
τ
2
sin
π
Tf
π
Tf
2
size 12{ lline H left (f right ) rline rSup { size 8{2} } = left ( { {T} over {τ} } right ) rSup { size 8{2} } left ( { {"sin"πital "Tf"} over {πital "Tf"} } right ) rSup { size 8{2} } } {}
The noise power o/p with white noise input
N
o
=
η
2
T
τ
2
∫
−
∞
∞
sin
π
Tf
π
Tf
2
df
=
ηT
2τ
2
size 12{N rSub { size 8{o} } = { {η} over {2} } left ( { {T} over {τ} } right ) rSup { size 8{2} } Int cSub { size 8{ - infinity } } cSup { size 8{ infinity } } { left ( { {"sin"πital "Tf"} over {πital "Tf"} } right ) rSup { size 8{2} } ital "df"= { {ηT} over {2τrSup { size 8{2} } } } } } {}
(The integral has a value =π)
Noise bandwidth:
If a real filter with transfer fn
H
f
size 12{H left (f right )} {} centered at
f
o
size 12{f rSub { size 8{o} } } {} is used, we can consider an equivalent rectangular filter centered at
f
o
size 12{f rSub { size 8{o} } } {} with a bandwidth
B
N
size 12{B rSub { size 8{N} } } {} passing the same noise power.
B
N
size 12{B rSub { size 8{N} } } {} is called the noise bandwidth of the real filter
For the RC Filtered curves, the area can be shown to be
N
o
RC
=
π
2
ηf
c
size 12{N rSub { size 8{o} } left ( ital "RC" right )= { {π} over {2} }ηf rSub { size 8{c} } } {}
For a rectangular filter we have
N
o
rect
=
η
2
2B
N
=
ηB
N
size 12{N rSub { size 8{o} } left ( ital "rect" right )= { {η} over {2} } 2B rSub { size 8{N} } =ηB rSub { size 8{N} } } {}
Setting
N
0
(
RC
)
=
N
0
(
rect
)
size 12{N rSub { size 8{0} } \( ital "RC" \) =N rSub { size 8{0} } \( ital "rect" \) } {} ,
B
N
=
π
2
f
c
size 12{B rSub { size 8{N} } = { {π} over {2} } f rSub { size 8{c} } } {}
Hence noise BW of RC filter is 1.57 times its 3 dB BW.
