---
title: "Hilbert transform; ideal LPF is non-causal"
subject: "Communication Systems"
chapter: 1
key: "semester_6/communication_systems:hilbert-ideal-lpf-noncausal"
source: https://www.bctengineer.com/q/semester_6/communication_systems/hilbert-ideal-lpf-noncausal
site: BCT Engineering
---

# Hilbert transform; ideal LPF is non-causal

Communication Systems · Chapter 1: Introduction

- Asked in: 2075 Bhadra (CS I), 2+6 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/hilbert-ideal-lpf-noncausal
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1

## Question

Define Hilbert transformation. Show that impulse response of an ideal low pass filter is non-causal.

## Short answer

Hilbert transform shifts all frequency components by -90 degrees for positive frequencies and +90 for negative, equivalent to convolution with 1/(pi t). An ideal low-pass filter has impulse response 2BK sinc[2B(t-td)], which is nonzero for t less than td, so the filter is non-causal.

## Answer

#### Hilbert transform
The **Hilbert transform** of a signal $x(t)$ is obtained by shifting the phase of every frequency component by $-90^\circ$ (positive frequencies) and $+90^\circ$ (negative frequencies), without changing the amplitudes:

$$
\hat{x}(t) = \frac{1}{\pi}\int_{-\infty}^{\infty} \frac{x(\tau)}{t-\tau}\,d\tau = x(t) * \frac{1}{\pi t}
$$

$$
\hat{X}(f) = -j\,\text{sgn}(f)\,X(f)
$$

Example: the Hilbert transform of $\cos 2\pi f_c t$ is $\sin 2\pi f_c t$. It is used in SSB generation (phase-shift method) and in analytic signal / pre-envelope representation.

#### Impulse response of an ideal LPF is non-causal
An ideal low-pass filter of bandwidth $B$, gain $K$ and delay $t_d$ has

$$
H(f) = \begin{cases} K\,e^{-j2\pi f t_d}, & |f| \le B \\ 0, & |f| > B \end{cases}
$$

Its impulse response is the inverse Fourier transform:

$$
\begin{aligned}
h(t) &= \int_{-B}^{B} K\,e^{-j2\pi f t_d}e^{j2\pi f t}\,df = K\int_{-B}^{B} e^{j2\pi f(t-t_d)}\,df \\
&= K\,\frac{e^{j2\pi B(t-t_d)} - e^{-j2\pi B(t-t_d)}}{j2\pi(t-t_d)} \\
&= K\,\frac{\sin 2\pi B(t-t_d)}{\pi(t-t_d)} = 2BK\,\text{sinc}\left[2B(t-t_d)\right]
\end{aligned}
$$

where $\text{sinc}(x) = \sin(\pi x)/(\pi x)$.

```text
             h(t)  peak 2BK at t = td
               |       *
               |      * *
     . .  .  . |  .  *   *  .  .   . .
 ----------------+----+-+----+-----------> t
   (non-zero     0    td
    for t < 0)
```

**Causality check:** a system is causal only if $h(t) = 0$ for all $t < 0$. The sinc function extends from $-\infty$ to $+\infty$; it is zero only at the discrete points $t = t_d + n/(2B)$, $n \ne 0$. For example at $t = -\frac{1}{4B}$, $h(t) \ne 0$ in general. However large the delay $t_d$ is made, the oscillating tails for $t < 0$ never vanish completely.

So $h(t) \ne 0$ for $t < 0$, i.e. the filter would respond before the impulse is applied. Hence the **ideal LPF is non-causal and physically unrealizable**. (This also follows from the Paley–Wiener criterion: a causal filter cannot have $|H(f)| = 0$ over a whole band.) Practical filters (Butterworth, Chebyshev) only approximate the ideal response; a large delay $t_d$ makes the truncated sinc a good approximation.

## Check yourself

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

1. The Hilbert transform of a signal:
   A. Shifts all frequency components by -90 degrees for positive and +90 degrees for negative frequencies, keeping amplitude
   B. Doubles the amplitude of each component
   C. Removes negative frequencies
   D. Delays the signal by 90 ms
   Answer: A. H(f) = -j sgn(f), a pure phase shift; cos becomes sin. (Key point: Hilbert transform)

2. The impulse response of an ideal LPF (bandwidth B, gain K, delay td) is:
   A. K delta(t - td)
   B. 2BK sinc[2B(t - td)]
   C. K e^{-t/td} u(t)
   D. K u(t - td)
   Answer: B. Inverse transform of a rectangular spectrum gives a sinc centred at td. (Key point: Impulse response of ideal LPF)

3. Why is the ideal LPF non-causal?
   A. The impulse response is zero for all t
   B. Its gain is infinite
   C. Its phase is not linear
   D. The sinc response is non-zero for t < 0 however large td is, so it responds before the input
   Answer: D. A causal system needs h(t) = 0 for t < 0; the sinc tails extend to -infinity. (Key point: Non-causality)

## Related in Communication Systems

- [Classification of systems; ideal LPF is non-causal](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aclassify-systems-ideal-lpf) — Communication Systems, chapter 1
- [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 usefulness; impulse response of LPF](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aimpulse-response-usefulness-lpf) — Communication Systems, chapter 1
- [Matched filter approximation using ideal LPF](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-ideal-lpf) — Communication Systems, chapter 9, asked 2×
- [Matched filter approximation using RC low pass filter](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amatched-filter-rc-approximation) — Communication Systems, chapter 9, asked 2×
- [Noise equivalent bandwidth of RC low pass filter](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aneb-rc-lowpass) — Communication Systems, chapter 9

## Related in other subjects

- [Impulse response of ideal low pass filter](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-impulse-response) — Signals and Systems, chapter 5, asked 5×
- [Impulse response of DT ideal low pass filter](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-discrete-time) — Signals and Systems, chapter 6
- [Convolution sum; ideal LPF is non-causal](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aconvolution-sum-ideal-lpf-noncausal) — Signals and Systems, chapter 5
- [Ideal LPF impulse and step response](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Aideal-lpf-impulse-step) — Signals and Systems, chapter 5, asked 5×