---
title: "Optimum detector impulse response in additive white noise"
subject: "Communication Systems"
chapter: 9
key: "semester_6/communication_systems:optimum-detector-impulse-response"
source: https://www.bctengineer.com/q/semester_6/communication_systems/optimum-detector-impulse-response
site: BCT Engineering
---

# Optimum detector impulse response in additive white noise

Communication Systems · Chapter 9: Noise in Communication Systems

- Asked in: 2076 Asoj (CS II), 2+6 marks; 2071 Chaitra (CS II), 1+6 marks (asked 2 times)
- Video: [Introduction to Matched Filter and its Impulse Response Derivation | R K Classes | Lec 84](https://www.youtube.com/watch?v=_37bj4e1Iso) — R K Classes
- 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-impulse-response
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-9

## Question

What do you mean by optimum detector? Find the impulse response of optimum detector in the presence of additive white noise.

## Short answer

An optimum detector minimizes error probability by maximizing the output SNR. In AWGN, it is the matched filter with impulse response h(t)=k s(T−t), a time-reversed and delayed version of the transmitted pulse. The maximum SNR at the sampling instant is 2E/N_0.

## 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 the optimum detector in additive white noise

**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 passes strongly the frequencies where the signal is strong.

## Check yourself

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

1. What criterion defines an optimum detector for a known pulse in AWGN?
   A. Maximum bandwidth of the receiving filter
   B. Minimum probability of error, achieved by maximising output SNR at the sampling instant
   C. Minimum filter delay with unity gain
   D. Maximum noise power at the output
   Answer: B. The optimum detector minimises error probability, which for a known pulse in white Gaussian noise means maximising the output SNR at the sampling instant. (Key point: Optimum detector goal)

2. Why is the Schwarz inequality used in the derivation?
   A. It removes the noise from the filter output
   B. It converts the pulse into its energy spectrum
   C. It gives an upper bound on the output SNR and shows when equality holds
   D. It proves the filter must be non-causal
   Answer: C. The Schwarz inequality bounds the numerator, giving SNR <= 2E/N0, with equality only when H(f) = k S*(f) e^(-j2 pi f T). (Key point: Schwarz inequality step)

3. What transfer function maximises the output SNR?
   A. H(f) = k S*(f) e^(-j2 pi f T)
   B. H(f) = k S(f) e^(j2 pi f T)
   C. H(f) = k / S(f)
   D. H(f) = k |S(f)|^2
   Answer: A. Equality in the Schwarz inequality needs H to be proportional to the conjugate of S(f)e^(j2 pi f T), i.e. k S*(f) e^(-j2 pi f T). (Key point: Optimum transfer function)

4. What is the impulse response of the optimum detector?
   A. h(t) = k s(t - T), the pulse simply delayed by T
   B. h(t) = k s(t), the pulse itself
   C. h(t) = k s(-t) with no delay
   D. h(t) = k s(T - t), the pulse reversed in time and delayed by T
   Answer: D. The conjugate gives time reversal s(-t) and the exponential gives a delay T, so h(t) = k s(T - t). (Key point: Impulse response result)

## Related in Communication Systems

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