Define Matched filter

A matched filter is a signal processing filter designed to maximize the signal-to-noise ratio (SNR) of a known signal in the presence of additive white Gaussian noise. It is used extensively in communications, radar, sonar, and other applications where the detection of a specific signal in noise is crucial.

### Key Characteristics and Definition:

1. **Purpose**: 
   - The primary purpose of a matched filter is to detect the presence of a known signal waveform \( s(t) \) within a received signal \( r(t) \) corrupted by noise \( n(t) \).

2. **Design Principle**:
   - A matched filter is designed such that its impulse response is the time-reversed and complex-conjugated version of the signal waveform \( s(t) \):
     \[ h(t) = s^*(-t), \]
     where \( s^*(t) \) denotes the complex conjugate of \( s(t) \).

3. **Operation**:
   - The received signal \( r(t) \) is passed through the matched filter \( h(t) \), resulting in an output \( y(t) \):
     \[ y(t) = \int_{-\infty}^{\infty} r(\tau) h(t - \tau) \, d\tau = \int_{-\infty}^{\infty} r(\tau) s^*(t - \tau) \, d\tau. \]
   - This output \( y(t) \) represents a measure of the correlation between the received signal and the reference signal \( s(t) \).

4. **Signal Detection**:
   - In signal detection applications, the presence of the signal \( s(t) \) is determined by comparing the output \( y(t) \) to a threshold value. If \( y(t) \) exceeds a certain threshold, it indicates that the signal \( s(t) \) is present in the received signal \( r(t) \).

5. **Optimality**:
   - The matched filter is optimal in the sense that it maximizes the SNR at its output for a given received signal-to-noise ratio (SNR) at its input. This optimality is derived from the Wiener-Kolmogorov filter theory in signal processing.

6. **Applications**:
   - **Communications**: Used in receiver design for demodulating signals in communications systems.
   - **Radar and Sonar**: Helps detect and distinguish echoes or signals from targets against background noise.
   - **Medical Imaging**: Employed in ultrasound imaging and radar-based medical diagnostics.
   - **Wireless Communications**: Used for symbol detection and equalization in wireless communication channels.

In summary, a matched filter is a critical tool in signal processing for detecting known signals in noise, leveraging the principle of maximizing SNR through correlation. Its design and application are fundamental in various fields where reliable signal detection and measurement are essential.