---
title: "Optimum detector; matched filter; autocorrelation"
subject: "Communication Systems"
chapter: 9
key: "semester_6/communication_systems:optimum-detector-matched-autocorrelation"
source: https://www.bctengineer.com/q/semester_6/communication_systems/optimum-detector-matched-autocorrelation
site: BCT Engineering
---

# Optimum detector; matched filter; autocorrelation

Communication Systems · Chapter 9: Noise in Communication Systems

- Asked in: 2074 Chaitra (CS II), 2+6+3 marks
- Video: [What is a Matched Filter?](https://www.youtube.com/watch?v=Ci-EjiMJo3I) — Iain Explains Signals, Systems, and Digital Comms
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/optimum-detector-matched-autocorrelation
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9

## Question

What do you mean by optimum detector? Show that the impulse response of the matched filter is reverse delayed version of the input signal. Explain auto correlation function.

## Short answer

An optimum detector minimizes error probability by maximizing output SNR. Matched filter impulse response is h(t)=k s(T-t), the time-reversed and delayed signal, derived via Schwarz inequality giving max SNR=2E/N₀. Autocorrelation R_X(τ) measures similarity of a signal with its delayed copy; R_X(0) equals average power.

## Answer

#### Optimum detector

An **optimum detector** is a receiver (filter followed by sampler and threshold device) that makes the decision with the **minimum probability of error**. For a known pulse in additive white Gaussian noise, this is achieved by the filter that **maximises the output signal-to-noise ratio at the sampling instant**; this optimum filter is the **matched filter**.

#### Impulse response of matched filter is reverse delayed input

**Set-up.** The received signal is $x(t)=s(t)+w(t)$, $0\le t\le T$, where $s(t)$ is a known pulse and $w(t)$ is white noise of two-sided PSD $N_0/2$. It passes through an LTI filter $h(t)\leftrightarrow H(f)$ and is sampled at $t=T$:

```text
 s(t)+w(t)    +--------+   y(t)    sample     y(T)
 ----------->|  h(t)  |---------->--/ ---->  decision
              +--------+           at t=T
```

Output $y(t)=s_o(t)+n_o(t)$ with

$$
s_o(T)=\int_{-\infty}^{\infty}H(f)S(f)e^{j2\pi fT}df,\qquad E[n_o^2]=\frac{N_0}{2}\int_{-\infty}^{\infty}\lvert H(f)\rvert^2df
$$

We want $H(f)$ that maximises the output peak signal-to-noise ratio

$$
\eta=\frac{\lvert s_o(T)\rvert^2}{E[n_o^2]}=\frac{\left\lvert\int H(f)S(f)e^{j2\pi fT}df\right\rvert^2}{\frac{N_0}{2}\int\lvert H(f)\rvert^2df}
$$

**Schwarz inequality:** $\left\lvert\int\phi_1\phi_2\,df\right\rvert^2\le\int\lvert\phi_1\rvert^2df\int\lvert\phi_2\rvert^2df$, with equality only when $\phi_1(f)=k\,\phi_2^*(f)$. Put $\phi_1=H(f)$ and $\phi_2=S(f)e^{j2\pi fT}$:

$$
\eta\le\frac{\int\lvert H\rvert^2df\int\lvert S\rvert^2df}{\frac{N_0}{2}\int\lvert H\rvert^2df}=\frac{2}{N_0}\int_{-\infty}^{\infty}\lvert S(f)\rvert^2df=\frac{2E}{N_0}
$$

(using Rayleigh's theorem, $E$ = pulse energy). The maximum is reached when

$$
H_{opt}(f)=k\,S^*(f)\,e^{-j2\pi fT}
$$

**Impulse response.** Take the inverse transform. For real $s(t)$, $S^*(f)=S(-f)$, which is the transform of $s(-t)$; the factor $e^{-j2\pi fT}$ is a delay of $T$:

$$
\begin{aligned}
h_{opt}(t) &= k\int_{-\infty}^{\infty}S^*(f)e^{-j2\pi fT}e^{j2\pi ft}df \\
&= k\int_{-\infty}^{\infty}S(-f)e^{-j2\pi f(T-t)}df \\
&= k\,s(T-t)
\end{aligned}
$$

So the optimum (matched) filter impulse response is the input pulse **reversed in time and delayed by $T$**. Choosing the sampling instant $T$ equal to the pulse duration makes $h(t)$ causal.

```text
   s(t)                      h(t) = s(T - t)
    |\                          /|
    | \                        / |
    |  \                      /  |
 ---+---+---> t          ---+---+---> t
    0   T                   0   T
```

**Results:** maximum SNR $=2E/N_0$ depends only on pulse energy, not on pulse shape; the filter's frequency response is the conjugate of the pulse spectrum, so it gives the most weight to the frequencies where the signal is strongest.

#### Autocorrelation function

The **autocorrelation (AC) function** measures how similar a signal is to a time-shifted copy of itself. For a power signal or a wide-sense stationary (WSS) random process $X(t)$:

$$
R_X(\tau)=\lim_{T\to\infty}\frac{1}{T}\int_{-T/2}^{T/2}x(t)\,x(t+\tau)\,dt \quad\text{or}\quad R_X(\tau)=E[X(t)X(t+\tau)]
$$

For an energy signal, $R_g(\tau)=\int_{-\infty}^{\infty}g(t)g(t+\tau)\,dt$.

Main properties: $R(-\tau)=R(\tau)$ (even); $R(0)=$ average power (energy); $\lvert R(\tau)\rvert\le R(0)$; $R(\tau)\leftrightarrow S(f)$ (Wiener–Khinchin). Example: white noise has $R(\tau)=\frac{N_0}{2}\delta(\tau)$. Note that the matched filter output for the signal is $s_o(t)=kR_s(t-T)$, the pulse autocorrelation shifted to $T$, which peaks at $t=T$ because $R_s$ is maximum at zero shift.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is an optimum detector for a known pulse in AWGN?
   A. A receiver that has the widest possible bandwidth
   B. A receiver that makes decisions with minimum probability of error, using the filter that maximises output SNR at the sampling instant
   C. A receiver that removes all noise from the signal
   D. A receiver that maximises the output signal power regardless of noise
   Answer: B. The optimum detector minimises the probability of error, which for a known pulse in AWGN is achieved by maximising the output SNR at the sampling instant: the matched filter. (Key point: Definition of optimum detector)

2. In the matched filter derivation, when does the Schwarz inequality hold with equality, giving maximum SNR?
   A. When H(f) = k S(f)
   B. When H(f) is constant for all frequencies
   C. When H(f) = k S*(f) e^(-j2πfT)
   D. When H(f) is the magnitude of S(f) only
   Answer: C. Equality needs phi1 = k phi2*, so H(f) = k S*(f) e^(-j2πfT), the conjugate of the signal spectrum with a delay. (Key point: Schwarz inequality equality condition)

3. What is the impulse response of the matched filter for a real pulse s(t) sampled at t = T?
   A. h(t) = k s(t)
   B. h(t) = k s(T - t)
   C. h(t) = k s(t - T)
   D. h(t) = k s(-t - T)
   Answer: B. S*(f) corresponds to s(-t) and e^(-j2πfT) delays by T, so h(t) = k s(T - t): reversed in time and delayed by T. (Key point: Matched filter impulse response)

4. Which statement about the autocorrelation function R(τ) is correct?
   A. R(τ) is odd and R(0) is zero
   B. R(τ) is even, R(0) equals average power (or energy) and |R(τ)| ≤ R(0)
   C. R(τ) is largest at large τ
   D. R(τ) and the PSD are unrelated
   Answer: B. R(-τ) = R(τ), R(0) is the power or energy and it is the maximum value; R(τ) and PSD form a Fourier pair (Wiener-Khinchin). (Key point: Autocorrelation function)

## Close variants set in other exams

- [Optimum detector; matched filter impulse response](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-impulse-response) — Communication Systems, chapter 9, asked 3×

## Related in Communication Systems

- [Matched filter impulse response is reverse delayed input](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-reverse-delayed) — Communication Systems, chapter 9, asked 2×
- [Matched filter; impulse response of optimum detector](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-optimum-detector) — Communication Systems, chapter 9
- [Noise equivalent bandwidth; optimum detector impulse response](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aneb-optimum-detector) — Communication Systems, chapter 9
- [Optimum detector impulse response in additive white noise](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aoptimum-detector-impulse-response) — Communication Systems, chapter 9, asked 2×
- [Optimum detector impulse response; Hilbert transform](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aoptimum-detector-impulse-response-hilbert) — Communication Systems, chapter 9
- [Impulse response of matched filter](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-awgn) — Communication Systems, chapter 9, asked 3×