Page 229 - Fundamentals of Radar Signal Processing

will be Y(Ω) = H(Ω)X(Ω), where X(Ω) is the spectrum of the waveform (and

thus, except for a phase shift due to an overall delay, of a received target echo).
Consider maximizing the SNR at a specific time T . The power of the signal
M
component of the output at that instant is

(4.4)

To determine the output noise power, consider the case where the interference

is white noise with power spectral density . The noise power spectral
density at the output of the receiver will be . The total output
noise power is then

(4.5)

and the SNR measured at time T is
M

(4.6)

Clearly, χ depends on the receiver frequency response. The choice of H(Ω)
that will maximize χ can be determined via the Schwarz inequality. One of many
forms of the Schwarz inequality is

(4.7)

with equality if and only if B(Ω) = α A*(Ω), with α any arbitrary constant.
Applying Eq. (4.7) to the numerator of Eq. (4.6) give the upper bound on SNR
as

(4.8)

The SNR is maximized when