This module describes the quadrature components of narrowband noise
Narrowband representation
When passed through a narrowband filter noise can also be represented in terms of quadrature components
n
(
t
)
=
n
c
t
cos
2πf
o
t
−
n
s
t
sin
2πf
o
t
size 12{n \( t \) =n rSub { size 8{c} } left (t right )"cos"2πf rSub { size 8{o} } t - n rSub { size 8{s} } left (t right )"sin"2πf rSub { size 8{o} } t} {}
Where
f
o
size 12{f rSub { size 8{o} } } {} corresponds to
k
=
K
size 12{k=K} {} and lies in the centre of the band.
Letting
f
o
=
KΔf
size 12{f rSub { size 8{o} } =KΔf} {} and using
2πf
o
t
−
2πKΔ
ft
=
0
size 12{2πf rSub { size 8{o} } t - 2πKΔ ital "ft"=0} {}
We add the above to the arguments in the equation 1.
n
t
=
lim
Δf
→
0
∑
k
=
1
∞
a
k
cos
2π
f
0
+
k
−
K
Δf
t
+
b
k
sin
2π
f
0
+
k
−
K
Δf
t
size 12{n left (t right )= {"lim"} cSub { size 8{Δf rightarrow 0} } Sum cSub { size 8{k=1} } cSup { size 8{ infinity } } { left lbrace a rSub { size 8{k} } "cos"2π left [f rSub { size 8{0} } + left (k - K right )Δf right ]t+b rSub { size 8{k} } "sin"2π left [f rSub { size 8{0} } + left (k - K right )Δf right ]t right rbrace } } {}
Using identities for the sine and cosine of sum of two angles, we get the equation in terms of
n
c
size 12{n rSub { size 8{c} } } {} and
n
s
size 12{n rSub { size 8{s} } } {} , where
n
c
t
=
lim
Δf
→
0
∑
k
=
1
∞
a
k
cos
2π
k
−
K
Δ
ft
+
b
k
sin
2π
k
−
K
Δ
ft
size 12{n rSub { size 8{c} } left (t right )= {"lim"} cSub { size 8{Δf rightarrow 0} } Sum cSub { size 8{k=1} } cSup { size 8{ infinity } } { left [a rSub { size 8{k} } "cos"2π left (k - K right )Δ ital "ft"+b rSub { size 8{k} } "sin"2π left (k - K right )Δ ital "ft" right ]} } {}
n
s
t
=
lim
Δf
→
0
∑
k
=
1
∞
a
k
sin
2π
k
−
K
Δ
ft
−
b
k
cos
2π
k
−
K
Δ
ft
size 12{n rSub { size 8{s} } left (t right )= {"lim"} cSub { size 8{Δf rightarrow 0} } Sum cSub { size 8{k=1} } cSup { size 8{ infinity } } { left [a rSub { size 8{k} } "sin"2π left (k - K right )Δ ital "ft" - b rSub { size 8{k} } "cos"2π left (k - K right )Δ ital "ft" right ]} } {}
n
c
size 12{n rSub { size 8{c} } } {} and
n
s
size 12{n rSub { size 8{s} } } {} are stationary random processes represented as superposition of components.
It can be shown that
n
c
size 12{n rSub { size 8{c} } } {} and
n
s
size 12{n rSub { size 8{s} } } {} are Gaussian, zero mean equal variance uncorrelated variables.
significance:
n
(
t
)
size 12{n \( t \) } {} of frequency
kΔf
size 12{kΔf} {} gives rise to
n
c
size 12{n rSub { size 8{c} } } {} and
n
s
size 12{n rSub { size 8{s} } } {} of frequency
f
−
f
o
size 12{f - f rSub { size 8{o} } } {} within band -B/2 to B/2 and thus change insignificantly during one
f
o
size 12{f rSub { size 8{o} } } {} cycle.
n
c
size 12{n rSub { size 8{c} } } {} and
n
s
size 12{n rSub { size 8{s} } } {} represent amplitude variations of two slowly changing quadrature phasor components, the complete phasor is
r
t
=
n
c
2
t
+
n
s
2
t
1
2
size 12{r left (t right )= left [n rSub { size 8{c} } rSup { size 8{2} } left (t right )+n rSub { size 8{s} } rSup { size 8{2} } left (t right ) right ] rSup { size 8{ { {1} over {2} } } } } {}
θ
t
=
tan
−
1
n
s
t
/
n
c
t
size 12{θ left (t right )="tan" rSup { size 8{ - 1} } left [ {n rSub { size 8{s} } left (t right )} slash {n rSub { size 8{c} } left (t right )} right ]} {}
The endpoint of r wanders randomly with passage of time.
Psd of orthogonal noise
Select spectral components corresponding to
k
=
K
+
λ
size 12{k=K+λ} {} and
k
=
K
−
λ
size 12{k=K - λ} {} where
λ
size 12{λ} {} is an integer.
k
=
K
size 12{k=K} {} corresponds to frequency
f
o
size 12{f rSub { size 8{o} } } {} , hence selected components correspond to
f
o
+
λΔf
size 12{f rSub { size 8{o} } +λΔf} {} and
f
o
−
λΔf
size 12{f rSub { size 8{o} } - λΔf} {} , and these frequencies generate 4 power spectral lines.
Select from
n
c
(
t
)
size 12{n rSub { size 8{c} } \( t \) } {} that part
Δn
c
(
t
)
size 12{Δn rSub { size 8{c} } \( t \) } {} corresponding to the frequencies we have selected above
Δn
c
t
=
a
K
−
λ
cos
2
πλ
Δ
ft
−
b
K
−
λ
sin
2
πλ
Δ
ft
+
a
K
+
λ
cos
2
πλ
Δ
ft
+
b
K
+
λ
sin
2
πλ
Δ
ft
size 12{Δn rSub { size 8{c} } left (t right )=a rSub { size 8{K - λ} } "cos"2 ital "πλ"Δ ital "ft" - b rSub { size 8{K - λ} } "sin"2 ital "πλ"Δ ital "ft"+a rSub { size 8{K+λ} } "cos"2 ital "πλ"Δ ital "ft"+b rSub { size 8{K+λ} } "sin"2 ital "πλ"Δ ital "ft"} {}
All 4 terms are at same frequency and represent uncorrelated processes. Then the power
P
λ
size 12{P rSub { size 8{λ} } } {} of
Δn
c
(
t
)
size 12{Δn rSub { size 8{c} } \( t \) } {} is the ensemble average of
Δn
c
(
t
)
2
size 12{ left [Δn rSub { size 8{c} } \( t \) right ] rSup { size 8{2} } } {} and this may be calculated at any time
t
1
size 12{t rSub { size 8{1} } } {} . Choosing
t
1
size 12{t rSub { size 8{1} } } {} such that
λΔ
ft
1
size 12{λΔ ital "ft" rSub { size 8{1} } } {} is an integer,
Δn
c
t
=
a
K
−
λ
+
a
K
+
λ
size 12{Δn rSub { size 8{c} } left (t right )=a rSub { size 8{K - λ} } +a rSub { size 8{K+λ} } } {}
P
λ
=
E
Δn
c
t
1
2
=
E
a
K
−
λ
+
a
K
+
λ
2
=
a
K
−
λ
2
¯
+
a
K
+
λ
2
¯
size 12{P rSub { size 8{λ} } =E left lbrace left [Δn rSub { size 8{c} } left (t rSub { size 8{1} } right ) right ] rSup { size 8{2} } right rbrace =E left [ left (a rSub { size 8{K - λ} } +a rSub { size 8{K+λ} } right ) rSup { size 8{2} } right ]= {overline {a rSub { size 8{ {} rSub { size 6{K - λ} } } } rSup {2} }} size 12{+ {overline {a rSub { {} rSub { size 6{K+λ} } } rSup {2} }} }} {}
We then find
P
λ
=
2G
nc
λΔf
Δf
=
2G
n
K
−
λ
Δf
Δf
=
2G
n
K
+
λ
Δf
Δf
size 12{P rSub { size 8{λ} } =2G rSub { size 8{ ital "nc"} } left (λΔf right )Δf=2G rSub { size 8{n} } left [ left (K - λ right )Δf right ]Δf=2G rSub { size 8{n} } left [ left (K+λ right )Δf right ]Δf} {}
So that
G
nc
λΔf
=
G
n
K
−
λ
Δf
=
G
n
K
+
λ
Δf
size 12{G rSub { size 8{ ital "nc"} } left (λΔf right )=G rSub { size 8{n} } left [ left (K - λ right )Δf right ]=G rSub { size 8{n} } left [ left (K+λ right )Δf right ]} {}
For continuous frequency variable, this becomes
G
nc
f
=
G
n
f
0
−
f
+
G
n
f
0
+
f
size 12{G rSub { size 8{ ital "nc"} } left (f right )=G rSub { size 8{n} } left (f rSub { size 8{0} } - f right )+G rSub { size 8{n} } left (f rSub { size 8{0} } +f right )} {}
Similar equation also results for
G
ns
size 12{G rSub { size 8{ ital "ns"} } } {} The psd's of the orthogonal components are shown below, and are obtained by shifting the +ve and -ve parts of plot of
G
n
size 12{G rSub { size 8{n} } } {} from
+
f
o
size 12{+f rSub { size 8{o} } } {} and
−
f
o
size 12{ - f rSub { size 8{o} } } {} to
x
=
0
size 12{x=0} {} and adding the displaced plots.
Thus the power of
n
(
t
)
size 12{n \( t \) } {} and the orthogonal components are equal
Example: For white noise filtered by a rectangular BPF centered at
f
o
size 12{f rSub { size 8{o} } } {} with
BW
=
B
size 12{ ital "BW"=B} {} ,
G
n
(
f
)
=
η
2
f
o
−
B
2
≤
∣
f
∣
≤
f
o
+
B
2
,
G
n
(
f
)
=
0
elsewhere
alignl { stack {
size 12{G rSub { size 8{n} } \( f \) = { {η} over {2} } ~f rSub { size 8{o} } - { {B} over {2} }<= lline f rline<= f rSub { size 8{o} } + { {B} over {2} } ,~} {} #
G rSub { size 8{n} } \( f \) =0~ ital "elsewhere" {}} } {}
Hence
G
n
(
f
o
+
f
)
=
G
n
(
f
o
−
f
)
size 12{G rSub { size 8{n} } \( f rSub { size 8{o} } +f \) =G rSub { size 8{n} } \( f rSub { size 8{o} } - f \) } {} and
G
nc
(
f
)
=
G
ns
(
f
)
=
G
n
(
f
o
−
f
)
+
G
n
(
f
o
+
f
)
=
η
2
+
η
2
=
η
∣
f
∣
≤
B
2
size 12{G rSub { size 8{ ital "nc"} } \( f \) =G rSub { size 8{ ital "ns"} } \( f \) =G rSub { size 8{n} } \( f rSub { size 8{o} } - f \) +G rSub { size 8{n} } \( f rSub { size 8{o} } +f \) = { {η} over {2} } + { {η} over {2} } =η~~ lline f rline<= { {B} over {2} } } {}
and are twice the magnitude of
G
n
(
f
o
+
f
)
size 12{G rSub { size 8{n} } \( f rSub { size 8{o} } +f \) } {} .
The power of the orthogonal components is:
σ
nc
2
=
σ
ns
2
=
∫
−
B
/
2
B
/
2
G
nc
(
f
)
df
=
ηB
size 12{σ rSub { size 8{ ital "nc"} } rSup { size 8{2} } =σ rSub { size 8{ ital "ns"} } rSup { size 8{2} } = Int rSub { size 8{ - {B} slash {2} } } rSup { size 8{ {B} slash {2} } } {G rSub { size 8{ ital "nc"} } } \( f \) ital "df"=ηB} {}
And that of
n
(
t
)
size 12{n \( t \) } {} is
σ
n
2
=
∫
−
fo
−
B
/
2
−
fo
+
B
/
2
G
n
(
f
)
df
+
∫
fo
−
B
/
2
fo
+
B
/
2
G
n
(
f
)
df
=
2
η
2
B
=
ηB
size 12{σ rSub { size 8{n} } rSup { size 8{2} } = Int rSub { size 8{ - ital "fo" - {B} slash {2} } } rSup { size 8{ - ital "fo"+ {B} slash {2} } } {G rSub { size 8{n} } } \( f \) ital "df"+ Int rSub { size 8{ ital "fo" - {B} slash {2} } } rSup { size 8{ ital "fo"+ {B} slash {2} } } {G rSub { size 8{n} } } \( f \) ital "df"=2 { {η} over {2} } B=ηB} {}
