Matched Filter

In any digital receiver, the fundamental goal at the sampling instant is to maximise the ratio of signal energy to noise power so that the probability of a decision error is minimised. The matched filter is the optimal linear filter that achieves the maximum possible output signal-to-noise ratio (SNR) at the sampling instant for a signal corrupted by additive white Gaussian noise (AWGN). It is a cornerstone concept in digital communications and radar, and appears regularly in GATE examinations.

Derivation Intuition: Why Time Reversal

The matched filter is derived by maximising the output SNR at a fixed sampling instant t = T. Using the Cauchy-Schwarz inequality applied to the convolution integral, it can be shown that the maximum SNR is achieved when the filter frequency response H(f) is proportional to the complex conjugate of the signal spectrum S*(f), with an additional linear phase term e^{-j2*pi*f*T} accounting for the sampling delay.

In the time domain, this optimal filter has an impulse response h(t) = s(T - t). This expression means the filter's impulse response is the time-reversed and delayed version of the signal s(t). For a real signal, this is simply the mirror image of s(t) flipped about the midpoint T/2 and shifted to start at t = 0.

The physical interpretation is that the matched filter correlates the received signal with a stored replica of the transmitted signal. This correlation accumulates signal energy coherently while averaging out uncorrelated noise, producing the maximum possible output SNR at the exact moment t = T.

Mathematical Expression

The peak output SNR at t = T for a matched filter receiving signal s(t) in AWGN with one-sided noise power spectral density N0 is:

SNR_max = 2E / N0, where E is the signal energy E = integral of s^2(t) dt from 0 to T.

This result is remarkable because the maximum SNR depends only on the signal energy E and the noise density N0, not on the shape of the signal. Two signals with the same energy but different waveforms achieve identical maximum SNR when each is processed by its own matched filter. This is why energy-per-bit Eb/N0 is the universal measure of digital communication performance.

The equivalent frequency-domain description is H(f) = k * S*(f) * exp(-j2*pi*f*T), where k is any positive constant. The output spectrum at t = T is proportional to |S(f)|^2, which concentrates the signal energy and whitens the effective noise spectrum through the correlator action.

Correlation Receiver Equivalence

The matched filter is mathematically equivalent to a correlation receiver that multiplies the received signal r(t) by the reference signal s(t) and integrates from 0 to T. Both implementations produce the same output sample at t = T with the same maximum SNR. The correlation receiver form is often easier to implement in digital hardware, while the matched filter form is more natural in analog and RF circuit design.

For binary signaling with two possible signals s0(t) and s1(t), the optimal receiver computes the correlation of r(t) with the difference signal [s1(t) - s0(t)] and compares it to a threshold. This is the basis for the optimal binary detector in AWGN.

Given:
Signal s(t) = rectangular pulse of amplitude A = 3 V over 0 to T = 1 microsecond
Noise PSD (one-sided) N0 = 2e-6 W/Hz

Why this formula applies:
For matched filter, peak output SNR = 2E/N0
E = A^2 * T for rectangular pulse

Formula:
SNR_max = 2 * E / N0
E = A^2 * T

Substitution:
E = (3)^2 * (1e-6) = 9 * 1e-6 = 9e-6 Joules
SNR_max = 2 * 9e-6 / 2e-6

Calculation:
SNR_max = 18e-6 / 2e-6 = 9

Final Answer:
Peak output SNR = 9 (dimensionless) = 9.54 dB
Note: This is independent of pulse shape; any 3V pulse over 1 microsecond with E=9e-6 J achieves this same SNR.

Exam Tip: The matched filter peak SNR formula is SNR = 2E/N0. GATE often gives A and T and asks for SNR. Compute E = A^2*T for rectangular pulses. Also remember h(t) = s(T-t) is the key impulse response formula that will appear in one-mark conceptual questions.

Quick Revision

• Matched filter maximises output SNR at the sampling instant t = T for a signal in AWGN.
• Impulse response: h(t) = s(T - t), which is the time-reversed and T-delayed version of the signal.
• Frequency response: H(f) = S*(f) * exp(-j2*pi*f*T), the conjugate of the signal spectrum with linear phase.
• Maximum output SNR: SNR_max = 2E/N0, where E is signal energy. Shape does not matter, only energy.
• Equivalent to a correlation receiver: integrating product of r(t) and s(t) over 0 to T gives identical result.
• GATE trap: SNR = 2E/N0 uses one-sided N0. If two-sided PSD N0/2 is given, formula becomes SNR = E/(N0/2) = 2E/N0 still, but be careful with factor-of-2 definitions in different textbooks.
• For a rectangular pulse of amplitude A and duration T, energy E = A^2 * T.