Digital Communications
MATCHED FILTER
Dr/ Amr Wageeh
Matched Filter
Threshold
ro(t)
𝑚
ෝ
ro(T)
• Filter that maximize instantaneous SNR at the output.
•
𝑆1𝑜 𝑇 −𝑆2𝑜 (𝑇) 2
2
SNR = ζ =
𝜎𝑛2
• Filter is matched to input signal
• Design on difference signal g(t) = S1(t)- S2(t)
SNR = ζ2 is max
hMF(t)
HMF(f)
ro(t)
ro(T)
Threshold
• Design on difference signal g(t) = S1(t)- S2(t)
SNR = ζ2 is max
g(t)+ N(t)
go(t)+ No(t)
hMF(t)
HMF(f)
Threshold
go(T)+ No(T)
Notes
• Dealing with filters: for deterministic signal
x(t)
h(t)
H(f)
y(t) = x(t)*h(t)
Y(f) = X(f).H(f)
y(t)
Notes
• Dealing with filters: for random signal
n(t)
Sn(f)
h(t)
no(t)
So(f)
H(f)
∞
2
Total power at output σ no= ‫׬‬−∞ So(f) 𝑑𝑓
Cauchy Shuartz inequlaity
∞
‫׬‬−∞ 𝑓1 (𝑥)𝑓2 (𝑥) 𝑑𝑥 2 ≤
∞
∞
2
‫׬‬−∞ 𝑓1 (𝑥) 𝑑𝑥 . ‫׬‬−∞ 𝑓2 (𝑥) 2 𝑑𝑥
if f1(x) = const f2*(x)
For Deterministic signal
• go(t) = F-1 {Go(f)}
=
∞
‫׬‬−∞ 𝐺𝑜 𝑓
𝑒 𝑗2𝜋𝑓𝑡 𝑑𝑓
∞
= ‫׬‬−∞ 𝐺 𝑓
𝐻(𝑓) 𝑒 𝑗2𝜋𝑓𝑡 𝑑𝑓
∞
go(T) = ‫׬‬−∞ 𝐺 𝑓
𝐻(𝑓) 𝑒 𝑗2𝜋𝑓𝑇 𝑑𝑓
For Random Signal
∞
• σ2n= ‫׬‬−∞ Sn(f) |𝐻(𝑓)|2 𝑑𝑓
• SNR
𝑔𝑜 (𝑇) 2
2
=ζ =
σ2n
∞
=
‫׬‬−∞ 𝐺 𝑓 𝐻(𝑓) 𝑒 𝑗2𝜋𝑓𝑇 𝑑𝑓 2
‫׬‬−∞ Sn(f) |𝐻(𝑓)|2 𝑑𝑓
∞
∞
=
‫׬‬−∞ 𝑆𝑛 (𝑓) 𝐻(𝑓)
‫׬‬−∞ Sn(f) |𝐻(𝑓)|2 𝑑𝑓
∞
‫׬‬−∞ Sn(f) |𝐻(𝑓)|2 𝑑𝑓
∞
≤
𝐺 𝑓
𝑒 𝑗2𝜋𝑓𝑇 𝑑𝑓 2
𝑆𝑛 (𝑓)
∞
‫׬‬−∞
𝐺 𝑓
𝑒 𝑗2𝜋𝑓𝑇 𝑑𝑓 2
𝑆𝑛 (𝑓)
Sn(f) |𝐻(𝑓)|2 𝑑𝑓
∞
= ‫׬‬−∞
𝐺 𝑓
𝑆𝑛 (𝑓)
∞
𝑒 𝑗2𝜋𝑓𝑇 2 df = ‫׬‬−∞
𝐺 𝑓
𝑆𝑛 (𝑓)
2 df
Matched filter design
∗
•
𝑆𝑛 (𝑓) 𝐻(𝑓) = const
𝐺 𝑓
𝑆𝑛 (𝑓)
𝑒 −𝑗2𝜋𝑓𝑇
∗
𝐺 𝑓 −𝑗2𝜋𝑓𝑇
• 𝐻𝑀𝐹 (𝑓) = const
𝑒
𝑆𝑛(𝑓)
∗
∗
[𝑆1 𝑓 −𝑆2 𝑓 ] −𝑗2𝜋𝑓𝑇
• 𝐻𝑀𝐹 (𝑓) = const
𝑒
𝑆𝑛(𝑓)
Summary in MF
• SNR = ζ2=
∞ 𝐺2 𝑓
‫׬‬−∞ 𝑆 (𝑓) df
𝑛
∗
∗
[𝑆1 𝑓 −𝑆2 𝑓 ] −𝑗2𝜋𝑓𝑇
• 𝐻𝑀𝐹 (𝑓) = const
𝑒
𝑆𝑛(𝑓)
ζ
𝑃𝑒 = 𝑄( )
2
Special case: in AWGN
• 𝑆𝑛 𝑓
𝑁𝑜
=
2
In case of binary Comm and equiprobable system
• ℎ𝑀𝐹 = 𝑆1 𝑇 − 𝑡 − 𝑆2 𝑇 − 𝑡 .
𝐸1 −𝐸2
• 𝑉𝑡ℎ =
2
• 𝑃𝑒 |𝑚𝑖𝑛 = 𝑄(
𝐸𝑔
2𝑁𝑜
)
Proof (cont)
• 𝑉𝑡ℎ =
𝑉𝑡ℎ =
𝐸1 − 𝐸2
2
𝑆1𝑜 𝑇 +𝑆2𝑜 (𝑇)
2
• 𝑆10 𝑡 = 𝑓 −1 𝑆10 𝑓
∞
= ‫׬‬−∞ 𝑆10 (𝑓)𝑒 𝑗2𝜋𝑓𝑡 𝑑𝑓
∞
= ‫׬‬−∞ 𝑆1 𝑓 𝐻𝑀𝐹 𝑓 𝑒 𝑗2𝜋𝑓𝑡 𝑑𝑓
∞
• 𝑆10 𝑇 = ‫׬‬−∞ 𝑆1 𝑓 𝐻𝑀𝐹 𝑓 𝑒 𝑗2𝜋𝑓𝑇 𝑑𝑓
∞
= ‫׬‬−∞ 𝑆1 𝑓 [ 𝑆1∗ 𝑓 − 𝑆2∗ 𝑓 ] 𝑒 −𝑗2𝜋𝑓𝑇 𝑒 𝑗2𝜋𝑓𝑇 𝑑𝑓
∞
∞
= ‫׬‬−∞ 𝑆1 (𝑓) 2 df - ‫׬‬−∞ 𝑆1 𝑓 𝑆2∗ 𝑓 df
= E1 – E12
By the same steps for S20 (T)
• 𝑆10 𝑇 = E12– E2
• 𝑉𝑡ℎ =
𝐸1 −𝐸2
2
Parallel realization of MF
• ℎ𝑀𝐹 = 𝑆1 𝑇 − 𝑡 − 𝑆2 𝑇 − 𝑡
r(t)
𝑆1 𝑇 − 𝑡
Threshold
=
𝑆2 𝑇 − 𝑡
𝐸1 − 𝐸2
2
Notes
𝑃𝑒 |= 𝑄(
𝐸𝑔
2𝑁𝑜
)
• 1. Prob of error depend on energy not waveform
𝑆1 𝑡
𝑆1 𝑡
1
T
t
t
T
𝑆2 𝑡
𝑆2 𝑡
t
T/2
t
T/4
3T/4
Notes
• Why MF is better ?
• MF benefits from entire Eg.
• Magnitude response of MF is equal to spectrum of signal.
• Choice of T
• 0 ≤ T ≤ Tb
• Causal: T must have MF causal
Optimizing Pe
• How to design S1(t) and S2(t).
• Cross correlation factor:
• 𝜌12 =
1
∞
‫𝑆)𝑡( 𝑆 ׬‬2 (𝑡)𝑑𝑡 =
𝐸 𝐸 −∞ 1
1 2
1
𝐸1 𝐸2
E12
• Represents the relation between signals S1(t) and S2(t).
• 𝑃𝑒 = Q(
𝐸1 +𝐸2 −2 𝐸1 𝐸2 𝜌12
2 𝑁𝑜
)
-1 ≤ 𝜌12 ≤ 1
𝑃𝑒 |= 𝑄(
𝐸𝑔
2𝑁𝑜
• Eg is the energy of g(t).
∞
• 𝐸𝑔 = ‫׬‬−∞ 𝑔(𝑡) 2 dt
∞
= ‫׬‬−∞ S1(t)−S2(t) 2 dt
∞
∞
∞
= ‫׬‬−∞ S1(t) 2 dt + ‫׬‬−∞ S2(t) 2 dt – 2 ‫׬‬−∞ S1(t)S2(t) dt
= E1 + E2 – 2 E12
)
Special cases for 𝜌12
• 𝜌12 = 1
fully correlated
• 𝜌12 = -1
antipodal signal
• 𝜌12 = 0
1. 𝜌12 = 1 fully correlated
• S1(t) = const S2(t)= +αS2(t).
Two signals have the same shape, polarity , with different amplitude
S1(t)
1
t
T
S2(t)
𝑃𝑒 = Q(
1/ α
t
T
𝐸1 +𝐸2 −2 𝐸1 𝐸2
2 𝑁𝑜
)= Q (
𝐸1 − 𝐸2
2 𝑁𝑜
)
2. 𝜌12 = -1 antipodal signal
• S1(t) = const S2(t)= -αS2(t).
Two signals have the same shape, different polarity
S1(t)
1
t
T
S2(t)
𝑃𝑒 = Q(
T
-α
t
𝐸1 +𝐸2 +2 𝐸1 𝐸2
2 𝑁𝑜
)= Q (
𝐸1 + 𝐸2
2 𝑁𝑜
)
3. 𝜌12 = 0
∞
‫׬‬−∞ 𝑆1 (𝑡)𝑆2 (𝑡)𝑑𝑡= 0
𝑃𝑒 = Q(
𝐸1 +𝐸2
)
2 𝑁𝑜
Correlator implementation
• Output of MF ro(T) can be implemented using correlator (fixed components).
𝑇𝑏
• 𝑟𝑜 (𝑇) = ‫𝑡𝑑 𝑡 𝑔 𝑡 𝑟 𝑜׬‬
r(t)
X
g(t)
𝑇𝑏
න
𝑜
𝑟𝑜 (𝑇)
Equivalent to
r(t)
g(T-t)
𝑟𝑜 (𝑡)
𝑟𝑜 (𝑇)
NWGN case (whitening filter)
n(t)
NWGN
PSD = Sn(f)
HW(f)
𝑛(𝑡)
෤
WGN
PSD = No/2
• 𝑆𝑛𝑜 𝑓 = 𝑆𝑛 𝑓 . 𝐻𝑊 (𝑓) 2
• No/2 = 𝑆𝑛 𝑓 . 𝐻𝑊 (𝑓) 2
• 𝐻𝑊 (𝑓) =
𝑁𝑜ൗ
2
𝑆ሚ1 𝑇 − 𝑡
𝑆𝑛 𝑓
r(t)
S(t)+n(t)
𝑆ሚ1(t)=S1(t)*hw(t)
HW(f)
𝑆ሚ (𝑡) + 𝑛(𝑡)
෤
𝑆ሚ2(t)=S2(t)*hw(t)
𝑆ሚ2 𝑇 − 𝑡
-
Proofs
Proof
• Note:
• X(-t)
X*(f)
• X(t-T)
X(f) e-j2πfT
∞
∞
∗
• E = ‫׬‬−∞ 𝑆1 (𝑡). 𝑆2 (𝑡) dt = ‫׬‬−∞ 𝑆1 (𝑓). 𝑆2 (𝑓) df cross energy (correlation between signals).
∗
∗
[𝑆 𝑓 −𝑆2 𝑓 ] −𝑗2𝜋𝑓𝑇
• 𝐻𝑀𝐹 (𝑓) = const 1
𝑒
𝑆𝑛(𝑓)
•
at AWGN
𝑆𝑛 𝑓 =
∗
𝑁𝑜
2
∗
[𝑆 𝑓 −𝑆 𝑓 ] −𝑗2𝜋𝑓𝑇
• 𝐻𝑀𝐹 (𝑓) = const 1 𝑁𝑜 2
𝑒
ൗ2
∗
𝑐𝑜𝑛𝑠𝑡
[𝑆
𝑁𝑜ൗ
1
2
∗
•
=
•
= S1(-(t-T) - S2(-(t-T)
𝑓 − 𝑆2
𝑓 ] 𝑒 −𝑗2𝜋𝑓𝑇
Proof (Cont)
ζ
• 𝑃𝑒 = 𝑄( )
2
• SNRo
= ζ2=
• Pe = Q(
1
2
∞ 𝐺2 𝑓
‫׬‬−∞ 𝑆 (𝑓) df
𝑛
2𝐸𝑔
𝑁𝑜
) = Q(
=
𝐸𝑔
2𝑁𝑜
)
2 ∞
2(𝑓) df = 2 E
𝐺
‫׬‬
𝑁𝑜 −∞
𝑁𝑜 g
Correlator implementation
• 𝑟𝑜 (𝑡) =r(t)*hMF(t)
= r(t)*g(T-t)
∞
= ‫׬‬−∞ 𝑟 𝜏 . 𝑔 𝑇 − 𝑡 − 𝜏 𝑑𝜏
∞
• 𝑟𝑜 (𝑡) = ‫׬‬−∞ 𝑟 𝜏 . 𝑔 𝑇 − 𝑇 − 𝜏
∞
∞
𝑑𝜏
= ‫׬‬−∞ 𝑟 𝜏 . 𝑔 𝜏 𝑑𝜏 = ‫׬‬−∞ 𝑟 𝑡 . 𝑔 𝑡 𝑑𝑡