Solution:

$$ y_0(t)=\int_{-\infty}^{\infty} y_{\text {in }}(\lambda) s\left(t_1-t+\lambda\right) d \lambda=R\left(t-t_1\right) $$

Equation above describes the output of the matched filter as the cross correlation between the input signal and a delayed replica of the transmitted signal.

This implies that the matched-filter the receiver can be replaced by a cross-correlation receiver that performs the same mathematical operation as shown in Fig.

The input signal y (t) is multiplied by a delayed replica of the transmitted signal s(t - Tr), and the product is passed through a low-pass filter to perform the integration.

The cross-correlation receiver of Fig. tests for the presence of a target at only a single time delay Tr. Targets at other time delays, or ranges, might be found by varying Tr. However, this requires a longer search time. The search time can be reduced by adding,

parallel channels, each containing a delay line corresponding to a particular value of Tr, as well as a multiplier and low-pass filter.

In some applications, it may be possible to record the signal on some storage medium and at a higher playback speed perform the search sequentially with different values of Tr. That is, the playback speed is increased in proportion to the number of time-delay intervals Tr that are to be tested.

Since the cross-correlation receiver and the matched-filter receiver are equivalent mathematically, the choice as to which one to use in a particular radar application is determined by which is more practical to implement.

The matched-filter receiver, or an approximation, has been generally preferred in the vast majority of applications.