Matched filters (Part 1 of 2) maximizes the output signal-to-noise ratio for a known radar signal at a predefined time. For comments please contact me at solo.hermelin@gmail.com. For more presentations on different subjects visit my website at http://www.solohermelin.com.

1. 1
Matched Filters and
Ambiguity Functions for
RADAR Signals
Part 1
SOLO HERMELIN
Updated: 01.12.08http://www.solohermelin.com

2. 2
SOLO
Matched Filters and Ambiguity Functions for RADAR Signals
Table of Content
RADAR RF Signals
Maximization of Signal-to-Noise Ratio
Continuous Linear Systems
The Matched Filter
The Matched Filter Approximations
1.Single RF Pulse
2. Linear FM Modulated Pulse (Chirp)
Discrete Linear Systems
RADAR Signals
Signal Duration and Bandwidth
Complex Representation of Bandpass Signals
Matched Filter Response to a Band Limited Radar Signal
Matched Filter Response to Phase Coding
Matched Filter Response to its Doppler-Shifted Signal

3. 3
SOLO
Matched Filters and Ambiguity Functions for RADAR Signals
Table of Content (continue – 1)
Ambiguity Function for RADAR Signals
Definition of Ambiguity Function
Ambiguity Function Properties
Cuts Through the Ambiguity Function
Ambiguity as a Measure of Range and Doppler Resolution
Ambiguity Function Close to Origin
Ambiguity Function for Single RF Pulse
Ambiguity Function for Linear FM Modulation Pulse
Ambiguity Function for a Coherent Pulse Train
Ambiguity Function Examples (Rihaczek, A.W.,
“Principles of High Resolution Radar”)
References
A
M
B
I
G
U
I
T
Y
F
U
N
C
T
I
O
N
S

4. 4
SOLO
The transmitted RADAR RF
Signal is:
( ) ( ) ( )[ ]ttftEtEt 0000 2cos ϕπ +=
E0 – amplitude of the signal
f0 – RF frequency of the signal
φ0 –phase of the signal (possible modulated)
The returned signal is delayed by the time that takes to signal to reach the target and to
return back to the receiver. Since the electromagnetic waves travel with the speed of light
c (much greater then RADAR and
Target velocities), the received signal
is delayed by
c
RR
td
21 +
≅
The received signal is: ( ) ( ) ( ) ( )[ ] ( )tnoisettttfttEtE dddr +−+−⋅−= ϕπα 00 2cos
To retrieve the range (and range-rate) information from the received signal the
transmitted signal must be modulated in Amplitude or/and Frequency or/and Phase.
ά < 1 represents the attenuation of the signal
RADAR Signal Processing
RADAR RF Signals

5. 5
SOLO
The received signal is:
( ) ( ) ( ) ( )[ ] ( )tnoisettttfttEtE dddr +−+−⋅−= ϕπα 00 2cos
( ) ( ) tRRtRtRRtR ⋅+=⋅+= 222111 & 
We want to compute the delay time td due to the time td1 it takes the EM-wave to reach
the target at a distance R1 (at t=0), from the transmitter, and to the time td2 it takes the
EM-wave to return to the receiver, at a distance R2 (at t=0) from the target. 21 ddd ttt +=
According to the Special Theory of Relativity
the EM wave will travel with a constant
velocity c (independent of the relative
velocities ).21 & RR 
The EM wave that reached the target at
time t was send at td1 ,therefore
( ) ( ) 111111 ddd tcttRRttR ⋅=−⋅+=−  ( )
1
11
1
Rc
tRR
ttd 

+
⋅+
=
In the same way the EM wave received from the target at time t was reflected at td2 ,
therefore
( ) ( ) 222222 ddd tcttRRttR ⋅=−⋅+=−  ( )
2
22
2
Rc
tRR
ttd 

+
⋅+
=
RADAR Signal Processing

6. 6
SOLO
The received signal is:
( ) ( ) ( ) ( )[ ] ( )tnoisettttfttEtE dddr +−+−⋅−= ϕπα 00 2cos
21 ddd ttt += ( )
1
11
1
Rc
tRR
ttd 

+
⋅+
= ( )
2
22
2
Rc
tRR
ttd 

+
⋅+
=
( ) ( )
2
22
1
11
21
Rc
tRR
Rc
tRR
tttttttt ddd 



+
⋅+
−
+
⋅+
−=−−=−






+
−
+
−
+





+
−
+
−
=−
2
2
2
2
1
1
1
1
2
1
2
1
Rc
R
t
Rc
Rc
Rc
R
t
Rc
Rc
tt d 



From which:
or:
Since in most applications we can
approximate where they appear in the arguments of E0 (t-td), φ (t-td),
however, because f0 is of order of 109
Hz=1 GHz, in radar applications, we must use:
cRR <<21, 
1,
2
2
1
1
≈
+
−
+
−
Rc
Rc
Rc
Rc




( )   





−⋅










++





−⋅










+=





−⋅





−+





−⋅





−⋅≈− 2
.
201
.
10
22
0
11
00
2
1
2
1
2
12
1
2
12
1
21
D
Ralong
FreqDoppler
DD
Ralong
FreqDoppler
Dd ttffttff
c
R
t
c
R
f
c
R
t
c
R
fttf

( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEtE ddDdr +−+−⋅+−= ˆˆˆ2cosˆ 00 ϕπα
where 21
2
2
1
121
2
02
1
01
ˆˆˆ,,,ˆˆˆ,
2ˆ,
2ˆ
dddddDDDDD ttt
c
R
t
c
R
tfff
c
R
ff
c
R
ff +=≈≈+=−≈−≈

Finally
Matched Filters in RADAR Systems
Doppler Effect

7. 7
SOLO
The received signal model:
( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEtE ddDdr +−+−⋅+−= ϕπα 00 2cos
Matched Filters in RADAR Systems
Delayed by two-
way trip time
Scaled down
Amplitude Possible phase
modulated
Corrupted
By noise
Doppler
effect
We want to estimate:
• delay td range c td/2
• amplitude reduction α
• Doppler frequency fD
• noise power n (relative to signal power)
• phase modulation φ

8. 8
Matched Filters in RADAR SystemsSOLO
α MV
R
EV
Target
Transmitter&
Receiver
The transmitted RADAR RF
Signal is:
( ) ( ) ( )[ ]ttftEtEt θπ += 00 2cos
( )
c
tR
td
02
≅
Since the received signal preserve the envelope shape of the known transmitted signal
we want to design a Matched Filter that will distinguish the signal from the receiver noise.
( ) ( )
λ
λ
0
/
0
0 22 0 tR
f
c
tR
f
fc
D

−=−≅
=
the received signal is: ( ) ( ) ( ) ( ) ( )[ ] ( )tnoisettttffttEtE ddDdr +−+−+−≈ θπα 00 2cos
Scaled Down
In Amplitude Two-Way
Delay
Possible
Phase Modulation
Doppler
Frequency
For R1 = R2 = R we obtain that
Return to Table of Content

9. 9
Matched Filters for RADAR Signals
( ) ( ) ( )tntstv ii
+=
Linear Filter
( )thopt
( ) ( ) ( )tntsty oo
+=
SOLO
Maximization of Signal-to-Noise Ratio
Consider the problem of choosing a linear time-invariant filter hopt (t) that maximizes
the output signal-to-noise ratio at a predefined time t0.
The input waveform is: ( ) ( ) ( )tntstv ii +=
( )tsi - a known signal component
( )tni - noise (stationary random process) component
The output waveform is: ( ) ( ) ( )tntsty oo
+=
Assume that the linear filter has a finite time memory T, then
( ) ( ) ( )∫ −=
T
iopto dtshts
0
00 τττ ( ) ( ) ( )∫ −=
T
iopto dtnhtn
0
00 τττ
The signal-to-noise ratio is defined as:
( )
( )0
2
0
2
tn
ts
N
S
o
o
=





To find hopt (t) a variational technique is applied, by defining a non-optimal filter
( ) ( ) ( )tgthth opt ε+= ( ) ( ) 0
0
0 =−∫
T
i dtsg τττwith: and ε any real.
( )0
2
tno - the mean square value of ( )0tno
Continuous Linear Systems

10. 10
Matched Filters for RADAR Signals
( ) ( ) ( )tntstv ii +=
Linear Filter
( ) ( )tgthopt ε+
( ) ( ) ( )tntsty oo
''' +=
SOLO
Maximization of Signal-to-Noise Ratio
The output signal s’o (t) and noise n’o (t) at time t0 are:
( ) ( ) ( )[ ] ( )
( ) ( ) ( ) ( ) ( )0
0
0
0
0
0
0
00'
tsdtsgdtsh
dtsghts
o
T
i
T
iopt
T
iopto
=−+−=
−+=
∫∫
∫
  
τττετττ
τττετ
( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )∫
∫∫∫
−+=
−+−=−+=
T
io
T
i
T
iopt
T
iopto
dtngtn
dtngdtnhdtnghtn
0
00
0
0
0
0
0
00'
τττε
τττεττττττετ
( )[ ] ( )[ ] ( ) ( ) ( ) ( ) ( )
2
0
0
2
0
00
2
0
2
0 2' 





−+−+= ∫∫
T
i
T
iooo dtngdtngtntntn τττετττε
By the definition of the optimal filter ( )[ ] ( )[ ]2
0
2
0' tntn oo ≥
Therefore ( ) ( ) ( ) ( ) ( ) 02
2
0
0
2
0
00 ≥





−+− ∫∫
T
i
T
io dtngdtngtn τττετττε
Continuous Linear Systems (continue – 1(

11. 11
Matched Filters for RADAR Signals
( ) ( ) ( )tntstv ii +=
Linear Filter
( ) ( )tgthopt ε+
( ) ( ) ( )tntsty oo
''' +=
SOLO
Maximization of Signal-to-Noise Ratio
This inequality is satisfied for all values of ε if and only if the first term vanishes
( ) ( ) ( ) ( ) ( ) 02
2
0
0
2
0
00 ≥





−+− ∫∫
T
i
T
io dtngdtngtn τττετττε
( ) ( ) 0
0
0 =−∫
T
i dtsg τττ
( ) ( ) ( ) 02
0
00 =−∫
T
io dtngtn τττ
Using we obtain:( ) ( ) ( )∫ −=
T
iopto dtnhtn
0
00 τττ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0
0 0
00
0 0
00 =−−=−− ∫∫∫∫
T T
iiopt
T T
iiopt ddtntnhgddtntnhg στστστστστστ
where is the Autocorrelation Function of the input noise.( ) ( ) ( )στστ −−=− 00: tntnR iinn ii
Continuous Linear Systems (continue – 2(

12. 12
Matched Filters for RADAR Signals
( ) ( ) ( )tntstv ii +=
Linear Filter
( ) ( )tgthopt ε+
( ) ( ) ( )tntsty oo
''' +=
SOLO
Maximization of Signal-to-Noise Ratio
Therefore the optimality condition is:
( ) ( ) ( ) 0
0 0
=





−∫ ∫ τσστστ ddRhg
T T
nnopt ii
( ) ( ) 0
0
0 =−∫
T
i dtsg τττComparing with the condition:
we obtain:
( ) ( ) ( ) TtskdRh i
T
nnopt ii
≤≤−=−∫ ττσστσ 00
0
k is obtained using:
( ) ( ) ( ) ( ) ( ) ( ) ( )
k
tn
ddRhh
k
dtshts o
T T
nnoptopt
T
iopto ii
0
2
0 00
00
1
=−=−= ∫∫∫ στστσττττ
( )
( )0
0
2
ts
tn
k
o
o
=
For T → ∞ we can take the Fourier Transfer of the result:
( ) ( ) ( )
( )
( )[ ]τσστσ −=





−∫∞→
0
0
0
2
0
lim ts
ts
tn
dRh i
o
o
T
nnopt
T ii
FF
Continuous Linear Systems (continue – 3(

13. 13
Matched Filters for RADAR SignalsSOLO
Maximization of Signal-to-Noise Ratio
( ) ( ) ( )
( )
( )[ ]τσστσ −=





−∫
∞
0
0
0
2
0
ts
ts
tn
dRh i
o
o
nnopt ii
FF
( ) ( ) ( )tntstv ii +=
Linear Filter
( )thopt
( ) ( ) ( )tntsty oo
+=
( ) ( ) ( )
( )
( ) 0*
0
0
2
tj
i
o
o
nnopt eS
ts
tn
H ii
ω
ωωω −
=Φ ( ) ( )
( )
( )
( )ω
ω
ω
ω
iinn
tj
i
o
o
opt
eS
ts
tn
H
Φ
=
− 0*
0
0
2
Continuous Linear Systems (continue – 4(
Return to Table of Content

14. 14
Matched Filters for RADAR Signals
( )tsi
t
T0 mt
SOLO
The Matched Filter
Assume that the two-sided noise spectrum density is of a white noise, i.e.
( ) ( )στδστ −=−
2
0N
R iinn
( ) ( )[ ] 2
0N
R iiii nnnn ==Φ τω F
then
( ) ( ) 0*2 tj
i
o
opt eS
N
k
H ω
ωω −
= ( ) ( ) Tttts
N
k
th i
o
opt ≤≤−= 0
2
0
( )tsi
t
t
( )tsi −
T0
0
T−
mt
( )tsi
t
t
t
( )tsi −
( ) ( ) Ttttts
N
k
th mmiopt ≤≤−= ,0
2
0
T0
0
T−
0 mtm
tT −
mt
The optimal filter, that maximizes the
Signal-to-Noise Ratio for a white noise is
called a Matched Filter of the known
Signal si (t).
We can see that for a known input
signal of finite duration T the optimal
Matched Filter is also of finite duration T.

15. 15
Matched Filters for RADAR Signals
( ) ( ) ( ) ( ) 0
2
0
0
2 tj
iiopt eS
N
k
SHS ω
ωωωω −
==
SOLO
The Matched Filter
The signal and the noise at the output of the matched filter are found as follows:
then
( ) ( ) ( )
( ) ( )
( )∫ ∫∫
+∞
∞−
+∞
∞−
−−
+∞
∞−
−






==
π
ω
ω
π
ω
ω ωωω
2
2
2
2 0*2 d
dvevseS
N
kd
eS
N
k
ts vj
i
ttj
i
o
ttj
i
o
o
m
( ) ( ) 0*2 tj
i
o
opt eS
N
k
H ω
ωω −
=
( ) ( ) ( )
( ) ( )∫∫ ∫
+∞
∞−
+∞
∞−
+∞
∞−
−−
+−== dvttvsvs
N
k
dv
d
eSvs
N
k
ii
o
vttj
ii
o
0
* 2
2
2 0
π
ω
ω ω
The Autocorrelation Function of the input signal is defined as: ( ) ( ) ( )∫
+∞
∞−
−= dvvsvsR iiss ii
ττ :
therefore: ( ) ( )0
2
ttR
N
k
ts iiss
o
o −=
( ) ( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
∗






=Φ= ωω
π
ωωωω
π
d
N
S
N
k
dHHtn o
i
o
optnnopto ii
2
2
2
1
2
1 2
2
2
( ) ( ) mtj
i
o
opt
eS
N
k
H ω
ωω −
=
*2
( ) ( ) ( )[ ]∫
+∞
∞−
== dvvs
N
k
R
N
k
ts i
o
ss
o
o ii
2
0
2
0
2

16. 16
Matched Filters for RADAR SignalsSOLO
The Matched Filter
therefore:
( ) ( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
∗






=Φ= ωω
π
ωωωω
π
d
N
S
N
k
dHHtn o
i
o
optnnopto ii
2
2
2
1
2
1 2
2
2
( ) ( ) ( )[ ]∫
+∞
∞−
== dvvs
N
k
R
N
k
ts i
o
ss
o
o ii
2
0
2
0
2
( )[ ]
( )
( )[ ]
( )
( )[ ]
( )∫
∫
∫
∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞− 





=












==





π
ω
ω
π
ω
ω
2222
2
2
2
2
2
2
2
2
2
2
2
0
d
S
N
dvvs
dN
S
N
k
dvvs
N
k
tn
ts
N
S
i
o
i
o
i
o
i
o
o
o
Max
Since by Parseval’s relation: (E – input signal energy)( )[ ] ( ) E
d
Sdvvs ii
== ∫∫
+∞
∞−
+∞
∞−
π
ω
ω
2
22
( )[ ]
( ) oo
o
Max N
E
tn
ts
N
S 2
2
2
0
==





We have: ( ) ( ) ( )∫
+∞
∞−
+−= dvttvsvs
N
k
ts ii
o
to 0
2
0
Independent of signal waveform

17. 17
Matched Filters for RADAR Signals
( ) ( )
( ) ( )


≤≤−=
= −∗
Ttttsth
eSH tj
00
0ω
ωω
SOLO
The Matched Filter (Summary(
s (t) - Signal waveform
S (ω) - Signal spectral density
h (t) - Filter impulse response
H (ω) - Filter transfer function
t0 - Time filter output is sampled (for Radar signals this is the time the received returned
signal is expected to arrive)
n (t) - noise
N (ω) - Noise spectral density
Matched Filter is a linear time-invariant filter hopt (t) that maximizes
the output signal-to-noise ratio at a predefined time t0, for a given signal s (t(.
The Matched Filter output is:
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) 0
00
tj
o eSSHSS
dttssdthsts
ω
ωωωωω
ττττττ
−∗
+∞
∞−
+∞
∞−
⋅=⋅=
+−=−= ∫∫
Return to Table of Content

18. 18
Matched Filters for RADAR SignalsSOLO
Matched Filter Output for White Noise Spectrum
s (t) - Signal waveform with energy E
S (ω) - Signal spectral density
h (t) - Filter impulse response
H (ω) - Filter transfer function
( ) ( ) ( ) ( ) ( ) ( )
( )0
* 0
2
1
2
1
ttRdeSSdeSHts ss
ttjtj
o −=== ∫∫
+∞
∞−
−
+∞
∞−
ωωω
π
ωωω
π
ωω
( ) ( ) ( ) ( )0*
'
0
2
ss
TheoremsParsevalT
RdSSdttsE === ∫∫
+∞
∞−
ωωω
so (t) - Filter output signal
N (ω) - Noise spectral density η/2
Rnn (τ) - Noise Autocorrelation Function η/2 δ (τ) ( ) ( ) ( )∫−
∞→
+=
T
T
T
nn dttntn
T
R ττ
1
lim
Rss (τ) - Signal Autocorrelation Function ( ) ( ) ( )∫−
∞→
+=
T
T
T
ss dttsts
T
R ττ
1
lim
S/N - Output Power signal-to-noise ratio E/(η/2)
t0 - Time filter output is sampled (for Radar signals this is the time the received returned
signal is expected to arrive)
Return to Table of Content

19. 19
Matched Filters for RADAR Signals
SOLO
The Matched Filter Approximations
1.Single RF Pulse
( )
( )




>
≤≤−
=
2/0
2/2/cos 0
p
pp
i
tt
ttttA
ts
ω
pt - pulse width
( ) ( )
( )
( )
( )
( )














−





 −
+
+





 +






=
= ∫−
−
2
2
sin
2
2
sin
2
cos
0
0
0
0
2/
2/
0
p
p
p
p
p
t
t
tj
i
t
t
t
t
tA
dtetAjS
p
p
ωω
ωω
ωω
ωω
ωω ω
Fourier Transform
0ω - carrier frequency
We found: ( ) ( ) ( ) ( )∫
+∞
∞−
−=−= dvtvsvs
N
k
ttR
N
k
ts ii
o
tss
o
to ii
22
00
0
therefore:
( ) ( )
( )
( )
( )[ ]
( )[ ]
( ) ( )[ ]
( ) ( )[ ]
( )
( ) ( ) ttt
N
tkA
ttt
N
kA
tttvttt
tttvttt
N
kA
ttdvtvt
ttdvtvt
N
kA
ttdvtvAvA
ttdvtvAvA
N
k
tR
N
k
ts
p
o
p
p
o
tt
p
tt
tp
p
t
ttp
o
p
tt
t
p
t
tt
o
tt
t
p
p
t
tt
o
ss
o
to
p
p
p
p
p
p
p
p
p
p
p
p
p
ii
0
2
0
2
1
1
2/
2/0
0
0
2/
2/0
0
02
2/
2/
00
2/
2/
00
2
2/
2/
00
2/
2/
00
0
cos/1cos
02sin
2
1
cos
02sin
2
1
cos
02coscos
02coscos
0coscos
0coscos
22
0
2
0
ωω
ω
ω
ω
ω
ω
ω
ωω
ωω
ωω
ωω
ω
−=−≈







<<−−++
<<−+−
=








<<−−+
<<−+
=








<<−−
<<−
==
<<
−
+
−
−
+
−
−
+
−
−
=
∫
∫
∫
∫
( ) ( ) ( )tntstv ii +=
Linear Filter
( )thopt
( ) ( ) ( )tntsty oo +=

20. 20
Matched Filters for RADAR SignalsSOLO
The Matched Filter Approximations
1.Single RF Pulse (continue – 1(
( )
( )




>
≤≤−
=
2/0
2/2/cos 0
p
pp
i
tt
ttttA
ts
ω
pt - pulse width
( ) ( )
( )
( )
( )
( )
0
0
2
2
sin
2
2
sin
2 0
0
0
0
*
tj
p
p
p
p
p
tj
iMF
e
t
t
t
t
tA
ejSjS
ω
ω
ωω
ωω
ωω
ωω
ωω
−
−














−





 −
+
+





 +






=
=
0ω - carrier frequency
We obtained:
( )
( )





≥
<−
==
p
pp
o
p
to
tt
ttttt
N
tkA
ts
0
cos/1 0
2
00
ω
( ) ( ) ( )tntstv ii +=
Linear Filter
( )thopt
( ) ( ) ( )tntsty oo +=
t
2
τ
2
τ
−
( )tso
0
2
N
Ak τ
ττ−
0=mt
Return to Table of Content

21. 21
Matched Filters for RADAR SignalsSOLO
The Matched Filter Approximations
1.Single RF Pulse (continue – 2(
( )
( )




>
≤≤−
=
2/0
2/2/cos 0
p
pp
i
tt
ttttA
ts
ω
pt - pulse width
( ) ( )
( )
( )
( )
( )
0
0
2
2
sin
2
2
sin
2 0
0
0
0
*
tj
p
p
p
p
p
tj
iMF
e
t
t
t
t
tA
ejSjS
ω
ω
ωω
ωω
ωω
ωω
ωω
−
−














−





 −
+
+





 +






=
=
0ω - carrier frequency
We obtained:
Return to Table of Content

22. 22
SOLO
2. Linear FM Modulated Pulse (Chirp)
( )
222
cos
2
0
pp
i
t
t
tt
tAts ≤≤−





+=
µ
ω
The Fourier Transform is:
( ) [ ]
( ) ( )∫∫
∫
−−
−












++−+












+−=
−





+=
2/
2/
2
0
2/
2/
2
0
2/
2/
2
0
2
exp
2
1
2
exp
2
1
exp
2
cos
p
p
p
p
p
p
t
t
t
t
t
t
i
dt
t
tjAdt
t
tjA
dttj
t
tAS
µ
ωω
µ
ωω
ω
µ
ωω
∫∫ −− 












 +
+−













 +
+













 −
−













 −
−=
2/
2/
2
0
2
0
2/
2/
2
0
2
0
2
exp
2
exp
22
exp
2
exp
2
p
p
p
p
t
t
t
t
dttjj
A
dttjj
A
µ
ωωµ
µ
ωω
µ
ωωµ
µ
ωω

Change variables: xt =




 −
−
µ
ωω
π
µ 0
yt =




 +
+
µ
ωω
π
µ 0
( ) ∫∫ −− 





−













 +
+



















 −
−=
2
1
2
1
2
exp
2
exp
22
exp
2
exp
2
2
2
0
2
2
0
Y
Y
X
X
i dt
y
jj
A
dt
x
jj
A
S
π
µ
ωωπ
µ
ωω
ω





 −
−=




 −
+=
µ
ωω
π
µ
µ
ωω
π
µ 0
2
0
1
2
&
2
pp t
X
t
X 




 +
−=




 +
+=
µ
ωω
π
µ
µ
ωω
π
µ 0
2
0
1
2
&
2
pp t
Y
t
Y
Define: ( )f
n
tf p ∆=−=∆ πωωµ
π
2
2
&
2
1
: 0
Matched Filters for RADAR Signals

23. 23
SOLO
2. Linear FM Modulated Pulse (continue – 1)
The Fourier Transform is:
( ) ( ) ( )
∫∫ −− 





−




 +
+











 −
−=
2
1
2
1
2
exp
2
exp
22
exp
2
exp
2
22
0
22
0
Y
Y
X
X
i dt
y
jj
A
dt
x
jj
A
S
π
µ
ωωπ
µ
ωω
ω
The first part gives the spectrum around ω = ω0, and the second part around ω = -ω0 :
where: are Fresnel Integrals,
which have the properties:
( ) ( ) ∫∫ ==
UU
dz
z
USdz
z
UC
0
2
0
2
2
sin&
2
cos
ππ
( ) ( ) ( ) ( )USUSUCUC −=−−=− &
( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( ) ( )[ ] ( ) ( )−+
++−=−+−




 −
+
+++




 −
−=
ωωωω
µ
ωω
µ
π
µ
ωω
µ
π
ω
002211
2
0
2211
2
0
2
exp
2
2
exp
2
ii
i
SSYSjYCYSjYCj
A
XSjXCXSjXCj
A
S
Matched Filters for RADAR Signals
( )
222
cos
2
0
pp
i
t
t
tt
tAts ≤≤−





+=
µ
ω
ωωωπωµ
π
∆=−∆=∆=∆
2
:&2:
2
1
: 0
n
ftf p

24. 24
SOLO Fresnel Integrals
Augustin Jean Fresnel
1788-1827
Define Fresnel Integrals
( ) ( )
( ) ( )
( ) ( )
( ) ( )∫ ∑
∑∫
∞
=
+
∞
=
+
+
−=





=
++
−=





=
α
α
αα
π
α
αα
π
α
0 0
14
2
0
34
0
2
!214
1
2
sin:
!1234
1
2
cos:
n
n
n
n
n
n
nn
x
dS
nn
x
dC
( ) ( )αααα
πα
SjCdj +=





∫0
2
2
exp
( ) ( ) 5.0±=∞±=∞± SC
( ) ( ) ( ) ( )USUSUCUC −=−−=− &
The Cornu Spiral is defined as the
plot of S (u) versus C (u)
duuSd
duuCd






=






=
2
2
2
sin
2
cos
π
π
( ) ( ) duSdCd =+
22
Therefore u may be thought as measuring arc
length along the spiral.

25. 25
SOLO
2. Linear FM Modulated Pulse (continue – 2)
The Fourier
Transform is:
Define:
( ) ( ) ( )[ ] ( ) ( )[ ]{ }2
21
2
210
2
XSXSXCXC
A
Si
+++=− +
µ
π
ωωAmplitude Term:
Square Law Phase Term: ( ) ( )
µ
ωω
ω
2
2
0
1
−
−=Φ
Residual Phase Term: ( ) ( ) ( )
( ) ( ) 4
1tan
5.05.0
5.05.0
tantan 11
1
21
211
2
π
ω
τ
==
+
+
→
+
+
=Φ −−
>>∆
−
f
XCXC
XSXS
( ) ( )n
t
fXn
t
fX
pp
−∆=+∆= 1
2
&1
2
21
( )ω2Φ( ) +
− ωω0iS
( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( ) ( )[ ] ( ) ( )−+
++−=−+−




 −
+
+++




 −
−=
ωωωω
µ
ωω
µ
π
µ
ωω
µ
π
ω
002211
2
0
2211
2
0
2
exp
2
2
exp
2
ii
i
SSYSjYCYSjYCj
A
XSjXCXSjXCj
A
S
Matched Filters for RADAR Signals
( )
222
cos
2
0
pp
i
t
t
tt
tAts ≤≤−





+=
µ
ω
ωωωπωµ
π
∆=−∆=∆=∆
2
:&2:
2
1
: 0
n
ftf p

26. 26
SOLO
2. Linear FM Modulated Pulse (continue – 3)
Matched Filters for RADAR Signals
( )
222
cos
2
0
pp
i
t
t
tt
tAts ≤≤−





+=
µ
ω
ωωωπωµ
π
∆=−∆=∆=∆
2
:&2:
2
1
: 0
n
ftf p
( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( ) ( )[ ] ( ) ( )−+
−
−−
++−=−+−




 −
+
+++




 −
−==
ωωωω
µ
ωω
µ
π
µ
ωω
µ
π
ωω
ω
ωω
002211
2
0
2211
2
0*
0
00
2
exp
2
2
exp
2
MFMF
tj
tjtj
iMF
SSeYSjYCYSjYCj
A
eXSjXCXSjXCj
A
eSS

27. 27
SOLO
2. Linear FM Modulated Pulse (continue – 4)
Matched Filters for RADAR Signals
( )
222
cos
2
0
pp
i
t
t
tt
tAts ≤≤−





+=
µ
ω ωωωπωµ
π
∆=−∆=∆=∆
2
:&2:
2
1
: 0
n
ftf p
The Matched Filter output is given by: ( ) ( ) ( ) ( )∫
+∞
∞−
−=−= dvtvsvs
N
k
ttR
N
k
ts ii
o
tss
o
to ii
22
00
0
( )
( ) ( )
( ) ( )








<<−




 −
+−





+
<<




 −
+−





+
=
∫
∫
+
−
+
−
2/
2/
2
0
2
0
2/
2/
2
0
2
0
0
2
cos
2
cos
0
2
cos
2
cos
2
0 p
p
p
p
tt
t
p
t
tt
p
o
to
ttdv
tv
tv
v
v
ttdv
tv
tv
v
v
N
k
ts
µ
ω
µ
ω
µ
ω
µ
ω
We discard the double frequency term, whose contribution to the value of integral
is small for large ω0,
( )
( )
( )








<<−











 −+
+−+





−+
<<











 −+
+−+





−+
=
∫
∫
+
−
+
−
2/
2/
22
0
2
0
2/
2/
22
0
2
0
2
0
2
22
2cos
2
cos
0
2
22
2cos
2
cos
0 p
p
p
p
tt
t
p
t
tt
p
o
to
ttdv
tvtv
tv
t
tvt
ttdv
tvtv
tv
t
tvt
N
Ak
ts
µµµ
ω
µ
µω
µµµ
ω
µ
µω
( )
( )















<<−
−+





 −+
+−
+






+−
<<
−+





 −+
+−
+






+−
=
+
−
+
−
+
−
+
−
0
242
2
22
2sin
2
sin
0
242
2
22
2sin
2
sin
2/
2/
0
22
0
2/
2/
2
0
2/
2/
0
22
0
2/
2/
2
0
2
tt
tv
tvtv
tv
t
tv
t
t
tt
tv
tvtv
tv
t
tv
t
t
N
Ak
p
tt
t
tt
t
p
t
tt
t
tt
o p
p
p
p
p
p
p
p
µµω
µµµω
µ
µµω
µµω
µµµω
µ
µµω
Expanding the integrand trigonometrically

28. 28
SOLO
2. Linear FM Modulated Pulse (continue – 5)
Matched Filters for RADAR Signals
Return to Table of Content
( )
222
cos
2
0
pp
i
t
t
tt
tAts ≤≤−





+=
µ
ω
ωωωπωµ
π
∆=−∆=∆=∆
2
:&2:
2
1
: 0
n
ftf p
The Matched Filter output is given by:
( )









<<−





+−
<<





+−
≈ +
−
+
−
0
2
sin
0
2
sin
2/
2/
2
0
2/
2/
2
0
2
0
tttv
t
t
tttv
t
t
tN
Ak
ts
p
tt
t
p
t
tt
o
to
p
p
p
p
µ
µ
ω
µ
µ
ω
µ
( )
( )






<<−





−−−





++−
<<





−+−−





+−
=
0
22
sin2/
2
sin
02/
2
sin
22
sin
2
0
2
0
2
0
2
02
tt
ttt
tttt
t
t
ttttt
t
t
ttt
t
tN
Ak
p
p
p
pp
p
o µµ
ωµ
µ
ω
µ
µ
ω
µµ
ω
µ
( ) ( )
( ) ( )
( )
( )
( )
( )









>
<














−






−
−
=







<<−





+
<<





−
=
p
p
p
p
p
p
p
o
p
pp
pp
o
tt
ttt
tt
tt
tt
tt
tt
N
tAk
ttttt
t
ttttt
t
tN
Ak
0
cos
/1
2
/1
2
sin
/1
2
0cos
2
sin2
0cos
2
sin2
0
2
0
02 ω
µ
µ
ω
µ
ω
µ
µ

29. 29
SOLO
2. Linear FM Modulated Pulse (continue – 6)
Matched Filters for RADAR Signals
Return to Table of Content
( )
222
cos
2
0
pp
i
t
t
tt
tAts ≤≤−





+=
µ
ω
ωωωπωµ
π
∆=−∆=∆=∆
2
:&2:
2
1
: 0
n
ftf p
The Matched Filter output is given by:
( )
( )
( )
( )
( )









>
<














−






−
−
≈
p
p
p
p
p
p
p
o
p
to
tt
ttt
tt
tt
tt
tt
tt
N
tAk
ts
0
cos
/1
2
/1
2
sin
/1 0
2
0
ω
µ
µ
o
p
N
tAk 2
pt
t
µ
π2
=∆
1>>ptµ

30. 30
SOLO
2. Linear FM Modulated Pulse (continue – 6)
Matched Filters for RADAR Signals
Return to Table of Content
( )
222
cos
2
0
pp
i
t
t
tt
tAts ≤≤−





+=
µ
ω
ωωωπωµ
π
∆=−∆=∆=∆
2
:&2:
2
1
: 0
n
ftf p

31. 31
Matched Filters for RADAR Signals
( ) ( ) ( )tntstv ii
+=
Linear Filter
( )Tnhopt
( ) ( ) ( )tntsty oo +=
( ) ( ) ( )TnnTnsTnv ii
+=
T T
( ) ( ) ( )TnnTnsTny oo
+=
SOLO
Maximization of Signal-to-Noise Ratio
Consider the problem of choosing a discrete linear time-invariant filter hopt (n T) that
Maximizes the discrete output signal-to-noise ratio at a predefined time mT.
The input waveform is: ( ) ( ) ( )tntstv ii +=
( )tsi - a known signal component
( )tni - noise (stationary random process) component
The output waveform is: ( ) ( ) ( )TnnTnsTny oo
+=
The signal-to-noise ratio at discrete time mT is defined as:
( )
( )Tmn
Tms
N
S
o
o
2
2
=





( )Tmno
2
- the mean square value of ( )Tmno
Discrete Linear Systems
The input and output of the discrete linear filter are synchronous discretized with
a constant time period T. S (z) is the Z-transform of the discrete signal input si (nT)
We have: ( ) ( ) ( )∫
+
−
=
σ
σ
ωωω
ω
σ
deeeTns TjTjTj
o
HS
2
1
( ){ } ( ) ( )∫
+
−
=
σ
σ
ω
ωω
σ
deTnnE Tj
o
22
2
1
HN

32. 32
Matched Filters for RADAR Signals
( ) ( ) ( )tntstv ii
+=
Linear Filter
( )Tnhopt
( ) ( ) ( )tntsty oo +=
( ) ( ) ( )TnnTnsTnv ii
+=
T T
( ) ( ) ( )TnnTnsTny oo
+=
SOLO
Maximization of Signal-to-Noise Ratio
Consider the problem of choosing a discrete linear time-invariant filter hopt (n T) that
Maximizes the discrete output signal-to-noise ratio at a predefined time mT.
Like in the continuous case the optimal H (z) is:
( )
( )
( ) ( )
( ) ( )∫
∫
+
−
+
−
==





σ
σ
ω
σ
σ
ω
ωω
σ
ωω
σ
de
de
Tmn
Tms
N
S
Tj
Tj
i
o
o
2
2
2
2
2
1
2
1
HN
HS
Discrete Linear Systems (continue – 1(
If N (ω) = N0 we have:
( ) ( ) ( )∫
+
−
=
σ
σ
ωωω
ω
σ
deeeTns TjTjTj
o
HSi
2
1
( ){ } ( ) ( )∫
+
−
=
σ
σ
ω
ωω
σ
deTnnE Tj
o
22
2
1
HN
( ) ( )
( )
mTj
Tj
iTj
e
e
ke ω
ω
ω
ω
−
=
N
S
H
( ) [ ] [ ]nms
N
k
nhz
zN
k
z i
m
i
−=⇔





= −
00
1
SH
Return to Table of Content

33. 33
RADAR SignalsSOLO
Waveforms
( ) ( ) ( )[ ]tttats θω += 0cos
a (t) – nonnegative function that represents any amplitude modulation (AM)
θ (t) – phase angle associated with any frequency modulation (FM)
ω0 – nominal carrier angular frequency ω0 = 2 π f0
f0 – nominal carrier frequency
Transmitted Signal
( ) ( ) ( )[ ]{ }ttjtats θω += 0exp
Phasor (complex) Transmitted Signal

34. 34
RADAR SignalsSOLO
Quadrature Form
( ) ( ) ( )[ ]
( ) ( )[ ] ( ) ( ) ( )[ ] ( )tttattta
tttats
00
0
sinsincoscos
cos
ωθωθ
θω
−=
+=
where: ( ) ( ) ( )[ ]
( ) ( ) ( )[ ]ttats
ttats
Q
I
θ
θ
sin
cos
=
=
( ) ( ) ( ) ( ) ( )ttsttsts QI 00 sincos ωω −=
One other form: ( ) ( ) ( )[ ] ( ) ( ) ( )
[ ]tjtjtjtj
ee
ta
tttats θωθω
θω −−+
+=+= 00
2
cos 0
( ) ( ) ( )[ ]tjtj
etgetgts 00 *
2
1 ωω −
+= ( ) ( ) ( ) ( ) ( )tj
QI etatsjtstg θ
=+=:
Envelope of the signal
( ) ( ) tj
etgts 0ω
=
Phasor (complex) Transmitted Signal

35. 35
RADAR SignalsSOLO
Spectrum
Define the Fourier Transfer F
( ) ( ){ } ( ) ( )∫
+∞
∞−
−== dttjtstsS ωω exp:F ( ) ( ){ } ( ) ( )∫
+∞
∞−
==
π
ω
ωωω
2
exp:
d
tjSSts -1
F
( ) ( ) ( )[ ]tjtj
etgetgts 00 *
2
1 ωω −
+= ( ) ( ) ( )[ ]0
*
0
2
1
ωωωωω −−+−= GGS-1
F
F
-1
F
F
( ) ( ) ( ) ( ) ( )tj
QI etatsjtstg θ
=+=:
( ) ( ) ( )[ ]tttats θω += 0cos
Inverse Fourier Transfer F -1
Envelope of the signalWe defined:

36. 36
RADAR SignalsSOLO
Energy ( ) ( ) ( )[ ]tttats θω += 0cos
( ) ( ) ( )[ ]{ } ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
≈++== dttadttttadttsEs
2
0
22
2
1
22cos1
2
1
: θω
Parseval’s Formula
Proof:
( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
= ωωω
π
dFFdttftf 2
*
12
*
1
2
1
( ) ( ) ( )∫
+∞
∞−
−= dttjtfF ωω exp11
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )∫∫ ∫∫ ∫∫
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
=−=−=
π
ω
ωω
π
ω
ωω
π
ω
ωω
22
exp
2
exp 2
*
112
*
2
*
12
*
1
d
FF
d
dttjtfFdt
d
tjFtfdttftf
( ) ( ) ( )∫
+∞
∞−
−=
π
ω
ωω
2
exp
*
2
*
2
d
tjFtf
If s (t) is real, than s (t) = s*(t) and
( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
=== ωω
π
dSdttsdttsEs
222
2
1
:

37. 37
RADAR SignalsSOLO
Energy (continue – 1) ( ) ( ) ( )[ ]tttats θω += 0cos
( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
=== ωω
π
dSdttsdttsEs
222
2
1
:
( ) ( ) ( ) ( )[ ] ( ) ( )[ ]
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) 





−−−+−−−+
−−−−+−−
=
−−+−−−+−=
−
−−
00
0000
0
*
0
*2
00
0
*
00
*
0
00
*
0
*
0
*
4
1
4
1
ϕϕ
ϕϕϕϕ
ωωωωωωωω
ωωωωωωωω
ωωωωωωωωωω
jj
jjjj
eGGeGG
GGGG
eGeGeGeGSS
For finite band (W << ω0 ) signals (see Figure)
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )∫∫∫
∫∫
∞+
∞−
∞+
∞−
∞+
∞−
+∞
∞−
−
+∞
∞−
=−−−−=−−
≈−−−=−−−
ωωωωωωωωωωωωω
ωωωωωωωωωω ϕϕ
dGGdGGdGG
deGGdeGG jj
*
0
*
00
*
0
2
0
*
0
*2
00 000
( ) ( ) gs EdGdSE 2
2
1
2
1
2
1
:
22
=≈= ∫∫
+∞
∞−
+∞
∞−
ωω
π
ωω
π
Return to Table of Content

38. 38
Signals
( ) ( )∫
+∞
∞−
= fdefSts tfi π2
SOLO
Signal Duration and Bandwidth
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )∫∫ ∫
∫ ∫∫ ∫∫
∞+
∞−
∞+
∞−
∞+
∞−
−
∞+
∞−
∞+
∞−
−
∞+
∞−
∞+
∞−
∞+
∞−
=







=








=







=
dffSfSdfdesfS
dfdefSsdfdefSsdss
tfi
tfitfi
ττ
τττττττ
π
ππ
2
22
( ) ( )∫
+∞
∞−
= fdefSts tfi π2 ( ) ( ) ( )∫
+∞
∞−
== fdefSfi
td
tsd
ts tfi π
π 2
2'
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )∫∫ ∫
∫ ∫∫ ∫∫
∞+
∞−
∞+
∞−
∞+
∞−
−
+∞
∞−
+∞
∞−
−
+∞
∞−
+∞
∞−
−
+∞
∞−
=







−=








−=







−=
dffSfSfdfdesfSfi
dfdesfSfidfdefSfsidss
tfi
tfitfi
222
22
2'2
'2'2''
πττπ
ττπττπτττ
π
ππ
( ) ( )∫∫
+∞
∞−
+∞
∞−
= dffSds
22
ττ
Parseval Theorem
From
From
( ) ( )∫∫
+∞
∞−
+∞
∞−
= dffSfdtts
2222
4' π

39. 39
Signals
( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )∫
∫
∫
∫ ∫
∫
∫ ∫
∫
∫
∫
∫
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
+∞
∞−
−
∞+
∞−
+∞
∞−
+∞
∞−
−
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
=====
dffS
fd
fd
fSd
fS
i
dffS
fdtdetstfS
dffS
tdfdefStst
dffS
tdtstst
tdts
tdtst
t
fifi
22
2
2
2
22
2
2
:
π
ππ
SOLO
Signal Duration and Bandwidth (continue – 1)
( ) ( )∫
+∞
∞−
−
= tdetsfS tfi π2
( ) ( )∫
+∞
∞−
= fdefSts tfi π2
Fourier
( ) ( )∫
+∞
∞−
−
−= tdetsti
fd
fSd tfi π
π 2
2
( ) ( )∫
+∞
∞−
= fdefSfi
td
tsd tfi π
π 2
2
( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )∫
∫
∫
∫ ∫
∫
∫ ∫
∫
∫
∫
∫
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
+∞
∞−
∞+
∞−
+∞
∞−
+∞
∞−
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
−
=








====
tdts
td
td
tsd
tsi
tdts
tdfdefSfts
tdts
fdtdetsfSf
tdts
fdfSfSf
fdfS
fdfSf
f
fifi
22
2
2
2
22
2
2222
:
ππ
ππππ

40. 40
Signals
( ) ( ) ( ) ( ) ( )∫∫∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
=≤








dffSfdttstdttsdttstdtts
222222
2
2
4'
4
1
π
( ) ( )∫∫
+∞
∞−
+∞
∞−
= dffSdts
22
τ
SOLO
Signal Duration and Bandwidth (continue – 2)
0&0 == ftChange time and frequency scale to get
From Schwarz Inequality: ( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
≤ dttgdttfdttgtf
22
Choose ( ) ( ) ( ) ( ) ( )ts
td
tsd
tgtsttf ':& ===
( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
≤ dttsdttstdttstst
22
''we obtain
( ) ( )∫
+∞
∞−
dttstst 'Integrate by parts
( )



=
+=
→



=
=
sv
dtstsdu
dtsdv
stu '
'
( ) ( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
∞+
∞−
+∞
∞−
−−= dttststdttsstdttstst '' 2
0
2

( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
−= dttsdttstst 2
2
1
'
( ) ( )∫∫
+∞
∞−
+∞
∞−
= dffSfdtts
2222
4' π
( )
( )
( )
( )
( )
( )
( )
( )∫
∫
∫
∫
∫
∫
∫
∫
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
=≤
dffS
dffSf
dtts
dttst
dtts
dffSf
dtts
dttst
2
222
2
2
2
222
2
2
44
4
1
ππ
assume ( ) 0lim =
→∞
tst
t

41. 41
SignalsSOLO
Signal Duration and Bandwidth (continue – 3)
( )
( )
( )
( )
( )
( )
    
22
2
222
2
2
4
4
1
ft
dffS
dffSf
dtts
dttst
∆
∞+
∞−
+∞
∞−
∆
∞+
∞−
+∞
∞−




























≤
∫
∫
∫
∫ π
Finally we obtain
( ) ( )ft ∆∆≤
2
1
0&0 == ftChange time and frequency scale to get
Since Schwarz Inequality: becomes an equality
if and only if g (t) = k f (t), then for:
( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
≤ dttgdttfdttgtf
22
( ) ( ) ( ) ( )tftsteAt
td
sd
tgeAts tt
ααα αα
222:
22
−=−=−==⇒= −−
we have ( ) ( )ft ∆∆=
2
1

42. 42
Signals
t
t∆2
t
( ) 2
ts
f
f
f∆2
( ) 2
fS
SOLO
Signal Duration and Bandwidth – Summary
then
( ) ( )∫
+∞
∞−
−
= tdetsfS tfi π2
( ) ( )∫
+∞
∞−
= fdefSts tfi π2
( ) ( )
( )
2/1
2
22
:














−
=∆
∫
∫
∞+
∞−
+∞
∞−
tdts
tdtstt
t
( )
( )∫
∫
∞+
∞−
+∞
∞−
=
tdts
tdtst
t
2
2
:
Signal Duration Signal Median
( ) ( )
( )
2/1
2
22
2
4
:














−
=∆
∫
∫
∞+
∞−
+∞
∞−
fdfS
fdfSff
f
π ( )
( )∫
∫
∞+
∞−
+∞
∞−
=
fdfS
fdfSf
f
2
2
2
:
π
Signal Bandwidth Frequency Median
Fourier
( ) ( )ft ∆∆≤
2
1
Return to Table of Content

43. 43
Matched Filters for RADAR Signals
( ) ( ) ( )[ ]tttats θω += 0cos
SOLO
Complex Representation of Bandpass Signals
The majority of radar signals are narrow band signals, whose Fourier transform is
limited to an angular-frequency bandwidth of W centered about a carrier angular
frequency of ±ω0.
Another form of s (t) is
( ) ( ) ( )
( )
( ) ( ) ( )
( )
( )
( ) ( ) ( ) ( )ttstts
tttatttats
QI
tsts QI
00
00
sincos
sinsincoscos
ωω
ωθωθ
−=
−=

sI (t) – in phase component sQ (t) – quadrature component
1
2
Define the signal complex envelope: ( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( )[ ]tjta
tjttatsjtstg QI
θ
θθ
exp
sincos:
=
+=+=
Therefore:
( ) ( ) ( )[ ] ( )[ ]tstjtgts ReexpRe 0 == ω
( ) ( ) ( ) ( ) ( ) ( ) ( )tststjtgtjtgts *
2
1
2
1
exp
2
1
exp
2
1
00 +=−+= ∗
ωω
or:
3
4
( ) ( ) ( )[ ]tjtjtats θω += 0exp
Analytic (complex) signal

44. 44
Matched Filters for RADAR Signals
( ) ( ) ( )[ ]tttats θω += 0cos
SOLO
Autocorrelation
The Autocorrelation Function is extensively used in Radar Signal Processing
( ) ( ) ( )∫
+∞
∞−
−= tdtstsRss ττ :
Real signalFor
The Autocorrelation Function is defined as:
Properties of the Autocorrelation Function:
2 ( ) ( )ττ ssss RR =−
( ) ( ) ( ) ( ) ( ) ( )ττττ
τ
ss
tt
ss RtdtststdtstsR =−=+=− ∫∫
+∞
∞−
+=+∞
∞−
'''
'
1 ( ) ( ) ( ) ( ) ( ) sss EfdfSfStdtstsR === ∫∫
+∞
∞−
+∞
∞−
*0 Es – signal energy
3
( ) ( ) ( ) ( ) ( ) ( )2222
2
2
0sss
EE
Inequality
Schwarz
ss REtdtstdtstdtstsR
ss
==−≤−= ∫∫∫
∞+
∞−
∞+
∞−
∞+
∞−
  
τττ
( ) ( )0ssss RR ≤τ
Autocorrelation is a mathematical tool for
finding specific patterns, such as the
presence of a known signal which has been
buried under noise.

45. 45
Matched Filters for RADAR SignalsSOLO
Autocorrelation (continue – 1(
The Autocorrelation Function is extensively used in Radar Signal Processing
( ) ( ) ( )∫
+∞
∞−
−= tdtgtgRgg ττ *:
Signal complex envelopeFor
The Autocorrelation Function is defined as:
Properties of the Autocorrelation Function:
2 ( ) ( )ττ *gggg RR =−
( ) ( ) ( ) ( ) ( ) ( )ττττ
τ
*''*'*
'
gg
tt
gg RtdtgtgtdtgtgR =−=+=− ∫∫
+∞
∞−
+=+∞
∞−
1 ( ) ( ) ( ) ( ) ( ) sgg EfdfGfGtdtgtgR 2**0 === ∫∫
+∞
∞−
+∞
∞−
Es – signal energy
3
( ) ( ) ( ) ( ) ( ) ( )22
2
2
2
2
2
2
04** ggs
EE
Inequality
Schwarz
gg REtdtgtdtgtdtgtgR
ss
==−≤−= ∫∫∫
∞+
∞−
∞+
∞−
∞+
∞−
  
τττ
( ) ( )0gggg RR ≤τ
( ) ( ) ( )[ ]tjtatg θexp:=

46. 46
Matched Filters for RADAR SignalsSOLO
Autocorrelation (continue – 2(
The Autocorrelation Function is extensively used in Radar Signal Processing
( ) ( ) ( )∫
+∞
∞−
−= tdtgtgRgg ττ *:
Signal complex envelopeFor
The Autocorrelation Function is defined as:
3
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( )
( ) ( ) ( ) ( )
( )
∫ ∫∫ ∫
∫ ∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
=
+∞
∞−
+∞
∞−
∂
∂
+
∂
∂
=






−−
∂
∂
==
∂
∂
=
    
0
11122
2
2
0
22211
1
1
0
212211
2
****
**00
gggg RR
gg
tdtgtgtdtg
t
tgtdtgtgtdtg
t
tg
tdtdtgtgtgtgR
τ
ττ
τ
τ
τ
( ) ( )0gggg RR ≤τ
( ) ( ) ( )[ ]tjtatg θexp:=
(continue – 1)
Since Rgg (0) is a maximum of a continuous function at τ=0, we must have
( ) 00
2
==
∂
∂
τ
τ
ggR
Therefore ( ) ( ) ( ) ( ) 0** =
∂
∂
+
∂
∂
∫∫
+∞
∞−
+∞
∞−
tdtg
t
tgtdtg
t
tg

47. 47
Matched Filters for RADAR Signals
( ) ( ) ( )[ ]tttats θω += 0cos
SOLO
Matched Filter for Received Radar Signals
The majority of radar signals are narrow band signals, whose Fourier transform is
limited to an angular-frequency bandwidth of W centered about a carrier angular
frequency of ±ω0.
The received signal will be:
1
• attenuated by a factor α
• retarded by a time t0 = 2 R/c
• affected by the Doppler effect
c
RR
c
f
c
D
 22
2 0
2
00
ω
λ
πω
ω
π
λ
−=−=
==
( ) ( ) ( )( ) ( )[ ]0000 cos ttttttats Dr −+−+−= θωωα2
Since the range and range-rate (t0, ωD) are not known exactly in advance,
the matched filter is designed to match the received signal at any time t0
assuming zero Doppler ωD=0.
Return to Table of Content

48. 48
Matched Filters for RADAR Signals
( ) ( ) ( ) ( ) ( )tjtgtjtgts 00 exp
2
1
exp
2
1
ωω −+= ∗
( ) ( )
( ) ( )


≤≤−=
= −∗
Ttttsth
eSH tj
00
0ω
ωω
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )[ ]∫
∫∫
∞+
∞−
∗∗
+∞
∞−
+∞
∞−






+−−+−++−+−





−+=
+−=−=
00000000
0
exp
2
1
exp
2
1
exp
2
1
exp
2
1
ttjttgttjttgjgjg
dttssdthstso
τωττωττωττωτ
ττττττ
SOLO
The Matched Filter is a linear time-invariant filter hopt (t) that maximizes
the output signal-to-noise ratio at a predefined time t0, for a known transmitted signal s (t(.
Assuming no Doppler let find the Matched Filter for the received radar signal at a time t0:
( ) ( ) ( ) ( ) ( )tjtgtjtgts 00 exp
2
1
exp
2
1
ωω −+= ∗
( )[ ] ( ) ( ) ( )[ ] ( ) ( )∫∫
+∞
∞−
∗
+∞
∞−
∗
+−−−++−−= τττωτττω dttggttjdttggttj 000000 exp
4
1
exp
4
1
( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )∫∫
+∞
∞−
∗
+∞
∞−
∗
+−−−+−+−−+ τωττωτωττω dtjttggttjdtjttggttj 00000000 2expexp
4
1
2expexp
4
1
Matched Filter Response to a Band Limited Radar Signal

49. 49
Matched Filter
output envelope
Matched Filters for RADAR Signals
( ) ( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
+−=−= ττττττ dttssdthstso 0
SOLO
Matched Filter Response to a Band Limited Radar Signal (continue – 1(
The transmitted radar signal:
( ) ( ) ( ) ( ) ( )tjtgtjtgts 00 exp
2
1
exp
2
1
ωω −+= ∗
( )[ ] ( ) ( ) ( )[ ] ( ) ( ) ( )






−+−−+






+−−= ∫∫
+∞
∞−
∗
+∞
∞−
∗
τωττωτττω dtjttggttjdttggttj 0000000 2expexpRe
2
1
expRe
2
1
The integral in the second term on the r.h.s. is the Fourier transform of
evaluated at ω = 2 ω0. Since the spectrum of is limited by ω = W << ω0, this
second term can be neglected, therefore:
( ) ( )[ ]0ttgg +−∗
ττ
( )τg
( ) ( )[ ] ( ) ( ) ( ) ( )[ ]tjtgdttggttjts o
filtermatchedsignal
o 0000 expRe
2
1
expRe
2
1
ωτττω =








+−−≈ ∫
∞+
∞−
∗

( ) [ ] ( ) ( ) [ ] ( )000000 exp
2
1
exp
2
1
ttRtjdttggtjtg gg
filtermatchedsignal
o −−=+−−= ∫
+∞
∞−
∗
ωτττω

( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( )[ ]tjta
tjttatsjtstg QI
θ
θθ
exp
sincos:
=
+=+=
Constant Phase
Matched Filter (for time t0) output is:
Autocorrelation Function of ( )tg Return to Table of Content

50. 50
Matched Filters for RADAR SignalsSOLO
Matched Filter Response to Phase Coding
( ) ( ) ( )


 ∆<<
=∆−= ∑
−
= elsewhere
tt
tftptfctg
M
p
p
0
011
0
Let the signal be a phase-modulated carrier, in which the modulation is in discrete and
equal steps Δt. The complex envelope of the signal can be described by a sequence of
complex numbers , such thatkc
( ) [ ] ( ) ( )∫
+∞
∞−
∗
+−−= dtttgtgtjgo 000exp
2
1
τωτ
Constant Phase
Matched Filter output envelope (change t ↔τ):
( )ttk ∆<≤+∆→ τττ 0
( ) [ ] ( ) ( )[ ]
[ ] ( )[ ]
( )
∑ ∫
∫∑
−
=
∆+
∆
∗
+∞
∞−
∗
−
=
∆−+−∆−=
∆−+−∆−∆−=+∆
1
0
1
0
1
0
0
exp
2
1
exp
2
1
M
p
tp
tp
p
M
p
po
dttkMtgctMj
dttkMtgtptfctMjtkg
τω
τωτ
Change variable of integration to t1 = t – τ + (M - k) Δt
( ) [ ] ( )
( )
( )
∑ ∫
−
=
−∆+−+
−∆−+
∗
∆−=+∆
1
0
1
110exp
2
1 M
p
tkMp
tkMp
po dttgctMjtkg
τ
τ
ωτ
tMt ∆=0 (expected receiving time)
( ) ( ) ( ) ( ) ( )tjtgtjtgts 00 exp
2
1
exp
2
1
ωω −+= ∗
The signal:

51. 51
Matched Filters for RADAR SignalsSOLO
Matched Filter Response to Phase Coding (continue – 1(
Matched Filter output envelope for a Phase Coding is:
( ) [ ] ( )[ ]
( )
∑ ∫
−
=
∆+
∆
∗
∆−+−∆−=+∆
1
0
1
0exp
2
1 M
p
tp
tp
po dttkMtgctMjtkg τωτ
Change variable of integration to t1 = t – τ + (M - k) Δt
( ) [ ] ( )
( )
( )
[ ] ( )
( )
( )
( )
( )
( )
∑ ∫∫∑ ∫
−
=
−∆+−+
∆−+
∗
∆−+
−∆−+
∗
−
=
−∆+−+
−∆−+
∗








+∆−=∆−=+∆
1
0
1
11110
1
0
1
110 exp
2
1
exp
2
1 M
p
tkMp
tkMp
tkMp
tkMp
p
M
p
tkMp
tkMp
po dttgdttgctMjdttgctMjtkg
τ
τ
τ
τ
ωωτ
( ) ( ) ( )
( ) ( ) ( ) τ
τ
−∆+−+<<∆−+=
∆−+<<−∆−+=
−+
∗
−−+
∗
tkMpttkMpctg
tkMpttkMpctg
kMp
kMp
11
*
1
11
*
1
( ) [ ] ∑ ∫∫
−
=
−∆
−+
−
−−+








+∆−=+∆
1
0 0
1
*
0
11
*
0exp
2
1 M
p
t
kMpkMppo dtcdtcctMjtkg
τ
τ
ωτ
( ) [ ] ∑
−
=
−+−−+ 











∆
−+





∆
∆−
∆
=+∆
1
0
*
1
*
0 1exp
2
1 M
p
kMpkMppo
t
c
t
cctMj
t
tkg
ττ
ωτ
This equation describes straight lines in the complex plane, that can have corners only at
τ = 0. At those corners
( ) [ ] ∑
−
=
−+∆−
∆
=∆
1
0
*
0exp
2
1 M
p
kMppo cctMj
t
tkg ω
Constant Phase

52. 52
Matched Filters for RADAR SignalsSOLO
Matched Filter Response to Phase Coding (continue – 2(
Matched Filter output envelope for a Phase Coding is:
( ) [ ] ∑
−
=
−+−−+ 











∆
−+





∆
∆−
∆
=+∆
1
0
*
1
*
0 1exp
2
1 M
p
kMpkMppo
t
c
t
cctMj
t
tkg
ττ
ωτ
This equation describes straight lines in the complex plane, that can have corners only at
τ = 0. At those corners
( ) [ ] ∑
−
=
−+∆−
∆
=∆
1
0
*
0exp
2
1 M
p
kMppo cctMj
t
tkg ω
Constant Phase
We can see that is the Discrete Autocorrelation Function for the
observation time t0 = M Δt (the time the received Radar signal return is expected)
∑
−
=
−+
1
0
*
M
p
kMpp cc

53. 53
Matched Filters for RADAR SignalsSOLO
Matched Filter Response to Phase Coding (continue – 3(
Example: Pulse poly-phase coded of length 4
Given the sequence: { } 1,,,1 −−++= jjck
which corresponds to the sequence of phases 0◦, 90◦, 270◦ and 180◦, the matched filter is
given in Figure bellow.
{ } 1,,,1
*
−+−+= jjck

54. 54
Pulse poly-phase coded of length 4
At the Receiver the coded pulse enters a 4 cells delay lane (from
left to right), a bin at each clock.
The signals in the cells are multiplied by -1,+j,-j or +1 and summed.
clock
SOLO
Poly-Phase Modulation
-1 = -11 1+
-j +j = 02 1+j+
+j -1-j = -13 1+j+j−
+1 +1+1+1 = 44 1+j+j−1−
-j-1+j = -1
5 j+j−1−
+j - j = 0
6
j−1−
7 1− -1 = -1
8 0
Σ
{ } 1,,,1 −−++= jjck
1− 1+j+ j− {ck*}
0 = 00
0
1
2
3
4
5
6
7
{ } 1,,,1* −+−+= jjck

55. 55
-1
Pulse bi-phase Barker coded of length 3
Digital Correlation
At the Receiver the coded pulse
enters a 3 cells delay lane (from left to
right), a bin at each clock.
The signals in the cells are multiplied
according to ck* sign and summed.
clock
-1 = -11
+1 -1 = 02
-( +1) = -15
0 = 06
+1 +1-( -1) = 33
+1-( +1) = 04
SOLO Pulse Compression Techniques
1
2
3
4
5
6
0
+1+1
0 = 00

56. 56
Pulse bi-phase Barker coded of length 5
Digital Correlation
At the Receiver the coded pulse enters a
7 cells delay lane (from left to right),
a bin at each clock.
The signals in the cells are multiplied
by ck* and summed.
clock
SOLO Pulse Compression Techniques
+1-1+1+1+1 { }*
kc
+1 = +11
+1 = 19
0 = 010
2 -1 +1 = 0
+1 +1 -1-( +1) = 04
+1 +1 +1 –(-1)+1 = 55
0 = 00
3 +1-1 +1 = 1
+1 +1 -(+1) -1 = 06
+1-( +1) +1 = 17
–(+1) +1 = 08
Return to Table of Content

57. 57
Matched Filters for RADAR Signals
( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( )[ ]tjta
tjttatsjtstg QI
θ
θθ
exp
sincos:
=
+=+=
SOLO
Matched Filter Response to its Doppler-Shifted Signal
Matched Filter for the transmitted radar signal:
The received radar signal has the form:
( ) ( ) ( )[ ]
( ) ( ) ( ) ( )tjtgtjtg
tttats
00
0
exp
2
1
exp
2
1
cos
ωω
θω
−+=
+=
∗
( ) ( ) ( ) ( )[ ]000 cos tttttakts Dr −++−= θωω
( ) ( ) ( ) ( )[ ]
( ) ( )[ ] ( ) ( ) ( )[ ] ( )tjtjtgtjtjtg
k
tttakts
DD
Dtr
0
*
0
00
expexp
2
1
expexp
2
cos
0
ωωωω
θωω
−+=
++==
( ) ( ) ( )∫
+∞
∞−
∗
=
−= τττ dtggtg
filtersignal
to 2
1
00
Matched Filter output envelope (designed under zero Doppler assumption) was found to be:
( ) ( ) ( ) ( )∫
+∞
∞−
∗
=
−= τττωτω dtgjgtg
filtersignal
DtDo   
exp
2
1
, 00
For a nonzero Doppler (ωD ≠ 0) the Matched Filter output envelope is:

58. 58
Matched Filters for RADAR Signals
( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( )[ ]tjta
tjttatsjtstg QI
θ
θθ
exp
sincos:
=
+=+=
SOLO
Matched Filter Response to its Doppler-Shifted Signal (continue – 1(
For a nonzero Doppler (ωD ≠ 0) the Matched Filter output complex envelope is:
( ) ( ) ( ) ( )∫
+∞
∞−
∗
=
−= τττωτω dtgjgtg
filtersignal
DtDo   
exp
2
1
, 00
Change between t and τ and define:
( ) ( ) ( ) ( )∫
+∞
∞−
∗
−= dttfjtgtgfX DD πττ 2exp:,
The magnitude of the complex envelope ,is called the Ambiguity Function.( )DfX ,τ
The name is sometimes used for , and sometimes even for .( )DfX ,τ ( ) 2
, DfX τ

59. 59
Matched Filters for RADAR SignalsSOLO
Matched Filter Response to its Doppler-Shifted Signal (continue – 2(
Properties of: ( ) ( ) ( ) ( )∫
+∞
∞−
∗
−= dttfjtgtgfX DD πττ 2exp:,
( ) ( ) ( ) ( ) ( ) ( )∫∫∫
+∞
∞−
∗
+∞
∞−
+∞
∞−
∗
=== dffGfGdttgdttgtgX
2
:0,01
2 ( ) ( ) ( )DDD fXfjfX ,*2exp, ττπτ =−−
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) [ ] ( ) ( )DDDD
tt
DD
DD
fXfjdttfjtgtgfj
dttfjtgtgfj
dttfjtgtgfX
,*2exp''2exp''*2exp
2exp2exp
2exp,
*
'
ττππττπ
τπττπ
πττ
τ
=






−=
+−+=
=−+=−−
∫
∫
∫
∞+
∞−
+=
∞+
∞−
∗
+∞
∞−
∗
( ) ( ) ( )DDD fXfjfX −−=− ,*2exp, ττπτ3
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( )[ ] ( ) ( )DDDD
tt
DD
DD
fXfjdttfjtgtgfj
dttfjtgtgfj
dttfjtgtgfX
−−=






−−−=
++−=
=+=−
∫
∫
∫
∞+
∞−
+=
∞+
∞−
∗
+∞
∞−
∗
,*2exp''2exp''*2exp
2exp2exp
2exp,
*
'
ττππττπ
τπττπ
πττ
τ

60. 60
Matched Filters for RADAR SignalsSOLO
Matched Filter Response to its Doppler-Shifted Signal (continue – 3(
Properties of: ( ) ( ) ( ) ( )∫
+∞
∞−
∗
−= dttfjtgtgfX DD πττ 2exp:,
4
5
( ) ( ) ( ) ( )τττ ggRdttgtgX =−= ∫
+∞
∞−
∗
:0,
( ) ( ) ( ) ( ) ( ) ( ) ( )fRdfffGfGdttfjtgtgfX GGDDD =+== ∫∫
+∞
∞−
+∞
∞−
∗ *
2exp,0 π
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )[ ] ( ) ( ) ( )[ ]
( ) ( ) ( )fRdfffGfG
dfdttffjtgfGdfdttffjtgfG
dttfjtgdftfjfGdttfjtgtgfX
GGD
DD
DDD
=+=






+−=+=
==
∫
∫ ∫∫ ∫
∫ ∫∫
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∞+
∞−
∗
+∞
∞−
∗
+∞
∞−
+∞
∞−
∗
*
*
2exp2exp
2exp2exp2exp,0
ππ
πππ
Return to Table of Content
( ) [ ] ( ) ( ) [ ] ( ) [ ] ( )0,exp
2
1
exp
2
1
exp
2
1
000000000 ttXtjttRtjdttggtjtg gg
filtermatchedsignal
o −−=−−=+−−= ∫
+∞
∞−
∗
ωωτττω

Autocorrelation Function of the
Signal Complex Envelope ( )tg
We found that the Matched Filter Output Complex Envelope is:( )tgo

61. 61
SOLO
Continue to
Ambiguity Functions
Matched Filters and Ambiguity Functions for RADAR Signals

62. January 18, 2015 62
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 –2013
Stanford University
1983 – 1986 PhD AA

63. 63
Fourier Transform
( ) ( ){ } ( ) ( )∫
+∞
∞−
−== dttjtftfF ωω exp:F
SOLO
Jean Baptiste Joseph
Fourier
1768-1830
F (ω) is known as Fourier Integral or Fourier Transform
and is in general complex
( ) ( ) ( ) ( ) ( )[ ]ωφωωωω jAFjFF expImRe =+=
Using the identities
( ) ( )t
d
tj δ
π
ω
ω =∫
+∞
∞− 2
exp
we can find the Inverse Fourier Transform ( ) ( ){ }ωFtf -1
F=
( ) ( ) ( ) ( ) ( )
( ) ( )( ) ( ) ( ) ( ) ( )[ ]00
2
1
2
exp
2
expexp
2
exp
++−=−=−=




−=
∫∫ ∫
∫ ∫∫
∞+
∞−
∞+
∞−
∞+
∞−
+∞
∞−
+∞
∞−
+∞
∞−
tftfdtfd
d
tjf
d
tjdjf
d
tjF
ττδττ
π
ω
τωτ
π
ω
ωττωτ
π
ω
ωω
( ) ( ){ } ( ) ( )∫
+∞
∞−
==
π
ω
ωωω
2
exp:
d
tjFFtf -1
F
( ) ( ) ( ) ( )[ ]00
2
1
++−=−∫
+∞
∞−
tftfdtf ττδτ
If f (t) is continuous at t, i.e. f (t-0) = f (t+0)
This is true if (sufficient not necessary)
f (t) and f ’ (t) are piecewise continue in every finite interval1
2 and converge, i.e. f (t) is absolute integrable in (-∞,∞)( )∫
+∞
∞−
dttf

64. 64
( )atf −
-1
F
F ( ) ( )ωω ajF −exp
Fourier TransformSOLO
( )tf
-1
F
F
( )ωFProperties of Fourier Transform (Summary)
Linearity1
( ) ( ){ } ( ) ( )[ ] ( ) ( ) ( )ωαωαωαααα 221122112211 exp: FFdttjtftftftf +=−+=+ ∫
+∞
∞−
F
Symmetry2
( )tF
-1
F
F
( )ωπ −f2
Conjugate Functions3 ( )tf *
-1
F
F
( )ω−*
F
Scaling4 ( )taf
-1
F
F






a
F
a
ω1
Derivatives5 ( ) ( )tftj
n
−
-1
F
F ( )ω
ω
F
d
d
n
n
( )tf
td
d
n
n
-1
F
F
( ) ( )ωω Fj
n
Convolution6
( ) ( )tftf 21
-1
F
F ( ) ( )ωω 21
* FF( ) ( ) ( ) ( )∫
+∞
∞−
−= τττ dtfftftf 2121
:*
-1
F
F ( ) ( )ωω 21
FF
( ) ( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
= ωωω dFFdttftf 2
*
12
*
1
Parseval’s Formula7
Shifting: for any a real8
( ) ( )tajtf exp
-1
F
F ( )aF −ω
Modulation9 ( ) ttf 0
cos ω -1
F
F
( ) ( )[ ]00
2
1
ωωωω −++ FF
( ) ( ) ( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
−=−= ωωω
π
ωωω
π
dFFdFFdttftf 212121
2
1
2
1

65. 65
Fourier Transform
( )tf
( ) ( )∑
∞
=
−=
0n
T Tntt δδ
( ) ( ) ( ) ( ) ( )∑
∞
=
−==
0
*
n
T
TntTnfttftf δδ
( )tf *
( )tf
T t
( ) ( ){ } ( ) σσ <== +∫
∞
−
f
ts
dtetftfsF
0
L
SOLO
Sampling and z-Transform
( ) ( ){ } ( ) σδδ <
−
==






−== −
∞
=
−
∞
=
∑∑ 0
1
1
00
sT
n
sTn
n
T
e
eTnttsS LL
( ) ( ){ }
( ) ( ) ( )
( ) ( ){ } ( ) ( )






<<
−
=
=






−
==
−
∞+
∞−
−−
∞
=
−
∞
=
+∫
∑∑
0
00
**
1
1
2
1
σσσξξ
π
δ
δ
ξ
σ
σ
ξ f
j
j
tsT
n
sTn
n
d
e
F
j
ttf
eTnfTntTnf
tfsF
L
L
L
( )
( ) ( )
( )
( )
( )
( )
( )
( )
( )
( )
( )
( )













−
=
−
−
=
−
=
∑∫
∑∫
∑
−−
−
−−
Γ
−−
−−
Γ
−−
∞
=
−
ts
e
ofPoles
tsts
F
ofPoles
tsts
n
nsT
e
F
Resd
e
F
j
e
F
Resd
e
F
j
eTnf
sF
ξ
ξξ
ξ
ξξ
ξ
ξ
ξ
π
ξ
ξ
ξ
π
1
1
0
*
112
1
112
1
2
1
Poles of
( ) Ts
e ξ−−
−1
1
Poles of
( )ξF
planes
T
nsn
π
ξ
2
+=
ωj
ωσ j+
0=s
Laplace Transforms
The signal f (t) is sampled at a time period T.
1Γ
2
Γ
∞→R
∞→R
Poles of
( ) Ts
e ξ−−
−1
1
Poles of
( )ξF
planeξ
T
nsn
π
ξ
2
+=
ωj
ωσ j+
0=s

66. 66
Fourier Transform
( )tf
( ) ( )∑
∞
=
−=
0n
T Tntt δδ
( ) ( ) ( ) ( ) ( )∑
∞
=
−==
0
*
n
T
TntTnfttftf δδ
( )tf *
( )tf
T t
SOLO
Sampling and z-Transform (continue – 1)
( ) ( )
( )
( )
( )
( ) ( ) ∑∑
∑∑
∞+
−∞=
∞+
−∞=
−−→
∞+
−∞=
−−
+→
+=
−
−−






+=
−






+
−=






+












−
−−
−=
−
−=
−−
−−
nn
Tse
n
ts
T
n
js
T
n
js
e
ofPoles
ts
T
n
jsF
TeT
T
n
jsF
T
n
jsF
e
T
n
js
e
F
RessF
ts
n
ts
π
π
π
π
ξ
ξ
ξ
ξπ
ξ
π
ξ
ξ
ξ
ξ
21
2
lim
2
1
2
lim
1
1
2
2
1
1
*
Poles of
( )ξF
ωj
σ
0=s
T
π2
T
π2
T
π2
Poles of
( )ξ*
F plane
js ωσ +=
The signal f (t) is sampled at a time period T.
The poles of are given by( )ts
e ξ−−
−1
1
( )
( )
T
n
jsnjTsee n
njTs π
ξπξπξ 2
21 2
+=⇒=−−⇒==−−
( ) ∑
+∞
−∞=






+=
n T
n
jsF
T
sF
π21*

67. 67
Fourier Transform
( )tf
( ) ( )∑
∞
=
−=
0n
T Tntt δδ
( ) ( ) ( ) ( ) ( )∑
∞
=
−==
0
*
n
T
TntTnfttftf δδ
( )tf *
( )tf
T t
SOLO
Sampling and z-Transform (continue – 2)
0=z
planez
Poles of
( )zF
C
The signal f (t) is sampled at a time period T.
The z-Transform is defined as:
( ){ } ( ) ( )
( )
( ) ( )
( )








−
−===
∑
∑
=
−
→
∞
=
−
=
iF
iF
i
iF
Ts
FofPoles
T
F
n
n
ze
ze
F
zTnf
zFsFtf
ξξ
ξ
ξ
ξξ
ξξξ
1
0
*
1
lim:Z
( )
( )





<
>≥
= ∫
−
00
0
2
1 1
n
RzndzzzF
jTnf
fC
C
n
π

68. 68
Fourier TransformSOLO
Sampling and z-Transform (continue – 3)
( ) ( ) ( )∑∑
∞
=
−
+∞
−∞=
=





+=
0
* 21
n
nsT
n
eTnf
T
n
jsF
T
sF
πWe found
The δ (t) function we have:
( ) 1=∫
+∞
∞−
dttδ ( ) ( ) ( )τδτ fdtttf =−∫
+∞
∞−
The following series is a periodic function: ( ) ( )∑ −=
n
Tnttd δ:
therefore it can be developed in a Fourier series:
( ) ( ) ∑∑ 





−=−=
n
n
n T
tn
jCTnttd πδ 2exp:
where: ( )
T
dt
T
tn
jt
T
C
T
T
n
1
2exp
1
2/
2/
=





= ∫
+
−
πδ
Therefore we obtain the following identity:
( )∑∑ −=





−
nn
TntT
T
tn
j δπ2exp
Second Way

69. 69
Fourier Transform
( ) ( ){ } ( ) ( )∫
+∞
∞−
−== dttjtftfF νπνπ 2exp:2 F
( ) ( ) ( )∑∑
∞
=
−
+∞
−∞=
=





+=
0
* 21
n
nsT
n
eTnf
T
n
jsF
T
sF
π
( ) ( ){ } ( ) ( )∫
+∞
∞−
== ννπνπνπ dtjFFtf 2exp2:2-1
F
SOLO
Sampling and z-Transform (continue – 4)
We found
Using the definition of the Fourier Transform and it’s inverse:
we obtain ( ) ( ) ( )∫
+∞
∞−
= ννπνπ dTnjFTnf 2exp2
( ) ( ) ( ) ( ) ( ) ( )∑∫∑
∞
=
+∞
∞−
∞
=
−=−=
0
111
0
*
exp2exp2exp
nn
n
sTndTnjFsTTnfsF ννπνπ
( ) ( ) ( )[ ]∫ ∑
+∞
∞−
+∞
−∞=
−−== 111
*
2exp22 νννπνπνπ dTnjFjsF
n
( ) ( ) ∑∫ ∑
+∞
−∞=
+∞
∞−
+∞
−∞=












−=





−−==
nn T
n
F
T
d
T
n
T
FjsF νπνννδνπνπ 2
11
22 111
*
We recovered (with –n instead of n) ( ) ∑
+∞
−∞=






+=
n T
n
jsF
T
sF
π21*
Second Way (continue)
Making use of the identity: with 1/T instead of T
and ν - ν 1 instead of t we obtain: ( )[ ] ∑∑ 





−−=−−
nn T
n
T
Tnj 11
1
2exp ννδννπ
( )∑∑ −=





−
nn
TntT
T
tn
j δπ2exp

70. 70
Fourier TransformSOLO
Henry Nyquist
1889 - 1976
http://en.wikipedia.org/wiki/Harry_Nyquist
Nyquist-Shannon Sampling Theorem
Claude Elwood Shannon
1916 – 2001
http://en.wikipedia.org/wiki/Claude_E._Shannon
The sampling theorem was implied by the work of Harry Nyquist in
1928 ("Certain topics in telegraph transmission theory"), in which
he showed that up to 2B independent pulse samples could be sent
through a system of bandwidth B; but he did not explicitly consider
the problem of sampling and reconstruction of continuous signals.
About the same time, Karl Küpfmüller showed a similar result, and
discussed the sinc-function impulse response of a band-limiting
filter, via its integral, the step response Integralsinus; this band-
limiting and reconstruction filter that is so central to the sampling
theorem is sometimes referred to as a Küpfmüller filter (but seldom
so in English).
The sampling theorem, essentially a dual of Nyquist's result,
was proved by Claude E. Shannon in 1949 ("Communication in
the presence of noise"). V. A. Kotelnikov published similar
results in 1933 ("On the transmission capacity of the 'ether' and
of cables in electrical communications", translation from the
Russian), as did the mathematician E. T. Whittaker in 1915
("Expansions of the Interpolation-Theory", "Theorie der
Kardinalfunktionen"), J. M. Whittaker in 1935 ("Interpolatory
function theory"), and Gabor in 1946 ("Theory of
communication").
http://en.wikipedia.org/wiki/Nyquist-Shannon_sampling_theorem

71. 71
SignalsSOLO
Signal Duration and Bandwidth
then
( ) ( )∫
+∞
∞−
−
= tdetsfS tfi π2
( ) ( )∫
+∞
∞−
= fdefSts tfi π2
t
t∆2
t
( ) 2
ts
f
f
f∆2
( ) 2
fS
( ) ( )
( )
2/1
2
22
:














−
=∆
∫
∫
∞+
∞−
+∞
∞−
tdts
tdtstt
t
( )
( )∫
∫
∞+
∞−
+∞
∞−
=
tdts
tdtst
t
2
2
:
Signal Duration Signal Median
( ) ( )
( )
2/1
2
22
2
4
:














−
=∆
∫
∫
∞+
∞−
+∞
∞−
fdfS
fdfSff
f
π ( )
( )∫
∫
∞+
∞−
+∞
∞−
=
fdfS
fdfSf
f
2
2
2
:
π
Signal Bandwidth Frequency Median
Fourier

72. 72
Signals
( ) ( )∫
+∞
∞−
= fdefSts tfi π2
SOLO
Signal Duration and Bandwidth (continue – 1)
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )∫∫ ∫
∫ ∫∫ ∫∫
∞+
∞−
∞+
∞−
∞+
∞−
−
∞+
∞−
∞+
∞−
−
∞+
∞−
∞+
∞−
∞+
∞−
=







=








=







=
dffSfSdfdesfS
dfdefSsdfdefSsdss
tfi
tfitfi
ττ
τττττττ
π
ππ
2
22
( ) ( )∫
+∞
∞−
= fdefSts tfi π2 ( ) ( ) ( )∫
+∞
∞−
== fdefSfi
td
tsd
ts tfi π
π 2
2'
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )∫∫ ∫
∫ ∫∫ ∫∫
∞+
∞−
∞+
∞−
∞+
∞−
−
+∞
∞−
+∞
∞−
−
+∞
∞−
+∞
∞−
−
+∞
∞−
=







−=








−=







−=
dffSfSfdfdesfSfi
dfdesfSfidfdefSfsidss
tfi
tfitfi
222
22
2'2
'2'2''
πττπ
ττπττπτττ
π
ππ
( ) ( )∫∫
+∞
∞−
+∞
∞−
= dffSds
22
ττ
Parseval Theorem
From
From
( ) ( )∫∫
+∞
∞−
+∞
∞−
= dffSfdtts
2222
4' π

73. 73
Signals
( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )∫
∫
∫
∫ ∫
∫
∫ ∫
∫
∫
∫
∫
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
+∞
∞−
−
∞+
∞−
+∞
∞−
+∞
∞−
−
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
=====
dffS
fd
fd
fSd
fS
i
dffS
fdtdetstfS
dffS
tdfdefStst
dffS
tdtstst
tdts
tdtst
t
fifi
22
2
2
2
22
2
2
:
π
ππ
SOLO
Signal Duration and Bandwidth
( ) ( )∫
+∞
∞−
−
= tdetsfS tfi π2
( ) ( )∫
+∞
∞−
= fdefSts tfi π2
Fourier
( ) ( )∫
+∞
∞−
−
−= tdetsti
fd
fSd tfi π
π 2
2
( ) ( )∫
+∞
∞−
= fdefSfi
td
tsd tfi π
π 2
2
( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
( )∫
∫
∫
∫ ∫
∫
∫ ∫
∫
∫
∫
∫
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
+∞
∞−
∞+
∞−
+∞
∞−
+∞
∞−
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
−
=








====
tdts
td
td
tsd
tsi
tdts
tdfdefSfts
tdts
fdtdetsfSf
tdts
fdfSfSf
fdfS
fdfSf
f
fifi
22
2
2
2
22
2
2222
:
ππ
ππππ

74. 74
Signals
( ) ( ) ( ) ( ) ( )∫∫∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
+∞
∞−
=≤








dffSfdttstdttsdttstdtts
222222
2
2
4'
4
1
π
( ) ( )∫∫
+∞
∞−
+∞
∞−
= dffSdts
22
τ
SOLO
Signal Duration and Bandwidth (continue – 1)
0&0 == ftChange time and frequency scale to get
From Schwarz Inequality: ( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
≤ dttgdttfdttgtf
22
Choose ( ) ( ) ( ) ( ) ( )ts
td
tsd
tgtsttf ':& ===
( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
≤ dttsdttstdttstst
22
''we obtain
( ) ( )∫
+∞
∞−
dttstst 'Integrate by parts
( )



=
+=
→



=
=
sv
dtstsdu
dtsdv
stu '
'
( ) ( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
∞+
∞−
+∞
∞−
−−= dttststdttsstdttstst '' 2
0
2

( ) ( ) ( )∫∫
+∞
∞−
+∞
∞−
−= dttsdttstst 2
2
1
'
( ) ( )∫∫
+∞
∞−
+∞
∞−
= dffSfdtts
2222
4' π
( )
( )
( )
( )
( )
( )
( )
( )∫
∫
∫
∫
∫
∫
∫
∫
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
∞+
∞−
+∞
∞−
=≤
dffS
dffSf
dtts
dttst
dtts
dffSf
dtts
dttst
2
222
2
2
2
222
2
2
44
4
1
ππ
assume ( ) 0lim =
→∞
tst
t

75. 75
SignalsSOLO
Signal Duration and Bandwidth (continue – 2)
( )
( )
( )
( )
( )
( )
    
22
2
222
2
2
4
4
1
ft
dffS
dffSf
dtts
dttst
∆
∞+
∞−
+∞
∞−
∆
∞+
∞−
+∞
∞−




























≤
∫
∫
∫
∫ π
Finally we obtain
( ) ( )ft ∆∆≤
2
1
0&0 == ftChange time and frequency scale to get
Since Schwarz Inequality: becomes an equality
if and only if g (t) = k f (t), then for:
( ) ( ) ( ) ( )∫∫∫
+∞
∞−
+∞
∞−
+∞
∞−
≤ dttgdttfdttgtf
22
( ) ( ) ( ) ( )tftsteAt
td
sd
tgeAts tt
ααα αα
222:
22
−=−=−==⇒= −−
we have ( ) ( )ft ∆∆=
2
1

76. 76
SOLO

77. 77
SOLO

78. 78
SOLO