Question

The matched filter is equivalent to which of the following?

• A. A correlator
• B. An envelope detector
• C. A PLL
• D. An equalizer

Hint:

In digital receiver design, remember this core equivalence: - A Matched Filter is a hardware implementation that uses an LTI filter with a specific impulse response. - A Correlator is a functional implementation that uses a multiplier followed by an integrator. Both approaches yield the exact same output value and maximum SNR at the sampling instant $t = T$.

Answer 1

The Correct Option is A

Understanding the Concept: A matched filter is an optimal linear filter designed to maximize the output signal-to-noise ratio (SNR) at a specific sampling moment in the presence of white Gaussian noise. The impulse response ($h(t)$) of a matched filter is a time-reversed and delayed version of the known clean input signal ($s(t)$): \[ h(t) = s(T - t) \] where $T$ is the total symbol duration.

Step 1: Finding the Filter Output using Convolution
The output $y(t)$ of any linear time-invariant (LTI) system is found by convolving its input signal $x(t)$ with its impulse response $h(t)$: \[ y(t) = x(t) * h(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) \, d\tau \] Now substitute the matched filter's characteristic impulse response, $h(t) = s(T - t)$, into this convolution equation: \[ y(t) = \int_{-\infty}^{\infty} x(\tau) s(T - (t - \tau)) \, d\tau = \int_{-\infty}^{\infty} x(\tau) s(T - t + \tau) \, d\tau \]

Step 2: Evaluating the Output at the Optimal Sampling Instant
We evaluate the filter's output at the optimal sampling moment, $t = T$: \[ y(T) = \int_{0}^{T} x(\tau) s(T - T + \tau) \, d\tau = \int_{0}^{T} x(\tau) s(\tau) \, d\tau \]

Step 3: Comparing with a Correlator Receiver
Let us look at how a standard correlator receiver operates:
• It multiplies the incoming noisy signal $x(\tau)$ directly by a locally generated clean copy of the signal $s(\tau)$.
• It integrates this product over the symbol time window from $0$ to $T$. The mathematical expression for this correlator operation is: \[ \text{Output}_{\text{correlator}} = \int_{0}^{T} x(\tau) s(\tau) \, d\tau \] Since the mathematical operations are identical, a matched filter sampled at $t = T$ is completely equivalent to a correlator receiver.

Step 4: Disproving alternative choices

• Envelope Detector: An asynchronous circuit built from a diode and capacitor used to extract the modulation envelope from AM signals, which does not maximize SNR using signal templates.
• PLL (Phase-Locked Loop): A closed-loop feedback control system used for tracking phase and frequency, not for optimal pulse detection.
• Equalizer: Used to reverse the distorting effects of a communication channel, rather than maximizing SNR for a known pulse shape in white noise.