1. White Noise Xt Has constant power SX( f ) at all frequencies.
In this case, we write SX( f ) = N0/2.
This corresponds to RX(τ) = (N0/2)δ(τ).
Having constant SX( f ) is an idealization. Actually, it is nearly constant for | f | up to GHZ and then tapers off.

2. However, what real systems see is |H( f )|2SX( f ),
where the bandwidth of a real system is below
1000 GHz. In other words, the hardware filters
the signal so that it does not matter what SX( f )
does for | f | > 1000 GHz.

4. τ

6. White Noise Through an RC FilterXt Yt

9. 10.7 The Matched Filter

10. received pulse
Pulse arrives with unknown delay.
Can you spot the pulse in the noise?

12. Output of Matched Filter Designed with t0 = 4.

14. Analysis
The output of the linear system is
...

15. 10.8 The Wiener Filter