Review of basic ideas from digital communication - Signal Processing and Optimization for Transceiver Systems - page 41

2.5.1 Case of white noise

For the special case where the noise is white, we have

S qq ( jω )= N 2 (2 . 51)

for all ω , as shown in Fig. 2.28. Since c is arbitrary, we can take the optimal
filter to be

B ( jω )= P ∗ ( jω ) . (2 . 52)

Taking the inverse Fourier transform, the impulse response of the filter is

β ( t )= p ∗ ( − t ) . (2 . 53)

For white noise, the matched filter β ( t ) is therefore the time-reversed conjugate of
the pulse p ( t ) . Figure 2.29 shows an example of a real pulse p ( t )anditsmatched
filter p (

− t ). Now, the signal component w ( t ) at the output of the filter β ( t )is

p )( t )= ∞
−∞

p ( τ ) p ∗ ( τ w ( t )=( β ∗ − t ) dτ, (2 . 54)

which is nothing but the deterministic autocorrelation of p ( t ). This is demon-
strated in Fig. 2.29(c). Note in particular that
w (0) = ∞

| 2 dτ =energyof p ( t ) .

−∞ |

p ( τ ) (2 . 55)

Thus, when the filter β ( t ) is matched to the pulse, the output at time zero is
the pulse energy. Moreover the maximized signal-to-noise ratio (2.48) becomes

= 2 ∞

∞
−∞ |

−∞ |p ( t ) | 2 dt
N 0

| 2

| w (0) 2

N 0

dω
2 π

= 2 E p
N 0

| 2 = P ( jω ) , (2 . 56)

σ e

where E p is the energy in the pulse p ( t ) .

Remarks

1. Autocorrelation of baseband pulse. For finite-duration p ( t ), even though the
duration of R pp ( τ ) is twice as long, the energy in the autocorrelation R pp ( τ )
is more narrowly concentrated, as demonstrated in Fig. 2.29(c). This is
the advantage of using a matched filter: it concentrates the signal energy
at the point where the detector samples the signal. For a given pulse
duration p ( t ), the achievable narrowness of the autocorrelation depends
on the bandwidth of the pulse; the larger the bandwith, the narrower the
autocorrelation can be. Thus, a baseband pulse p ( t )with large energy and
large bandwidth results in better performace at the output of the matched
filter, and, furthermore, adjacent pulses can be spaced closer (because of
the narrowness of the autocorrelation). Of course, bandwidth and energy
are valuable resources which cannot be increased arbitrarily.