Matched Filter Properties

© 2021 Institute for Communications Engineering, Technical University of Munich
Authors: Carolin Mirschina & Tasnad Kernetzky
Input pulse
$A_g = 1.00$
$\Delta t_g = 1.00$
$\tau_g = 0.00$
AWGN noise
$N_0 = 0.01$
$t$
−3
−2
−1
1
2
3
0
0.25
0.5
0.75
1
1.25
1.5
$g(t)$
$h(t)$

$t$
−3
−2
−1
1
2
3
0
0.25
0.5
0.75
1
1.25
1.5
$d_\rm{S}(t)$
Reception filter
$A_h = 1.00$
$\Delta t_h = 1.00$
$\tau_h = 0.00$
$T_\rm{D} = 0.00$
$\rm{Numerical \ results}$
$\rm{Variation}$$ \ \Delta t_g$ / $\Delta t_h$
$t$
−3
−2
−1
1
2
3
0
0.25
0.5
0.75
1
1.25
1.5
$h(t)^2$
$f$
−3
−2
−1
1
2
3
0
0.25
0.5
0.75
1
1.25
1.5
$|H(f)|^2$
$\rm{Variation} \ $$ \Delta t_g$
$\rm{Variation}\ $$ \Delta t_h$
$\rm{Step} \ = $
$K = $
5
10
15
20
25
0.60
0.70
0.80
0.90
1.00
1.10
1.20
1.30
1.40
$10 \cdot \lg \ [\rho_d(T_\rm{D}, \rm{opt})]$
$20 \cdot \lg \ [ K \cdot d_\rm{S}(T_\rm{D}, \rm{opt})]$
$20 \cdot \lg \ [K \cdot \sigma_d]$
$\rm{Pulse \ energy:}$
$E_g=1.0000$
$\rm{Useful \ detection \ sample:}$
$d_\rm{S}(T_\rm{D})=1.0000$
$\rm{Noise \ variance:}$
$\sigma_d^2=0.0050$
$\rho_d = d_\rm{S}^2(T_\rm{D}) / \sigma_d^2 =200.0$
$10 \cdot \lg \ \rho_d = 23.0\ \rm{dB}$
$10 \cdot \lg \ \rho_\rm{MF} = 23.0\ \rm{dB}$
Exercises

* First, select the number  $(1,\ 2, \text{...} \ )$  of the task to be processed.  The number  $0$  corresponds to a "Reset":  Same setting as at program start.
* A task description is displayed.  The parameter values are adjusted.  Solution after pressing "Show Solution".
* Both the input signal  $x(t)$  and the filter impulse response  $h(t)$  are normalized, dimensionless and energy-limited ("time-limited pulses").
* All times, frequencies, and power values are to be understood normalized, too.