---
title: "Ideal band pass and low pass filters (DT)"
subject: "Signals and Systems"
chapter: 6
key: "semester_4/signals_and_systems:ideal-bandpass-lowpass-dt"
source: https://www.bctengineer.com/q/semester_4/signals_and_systems/ideal-bandpass-lowpass-dt
site: BCT Engineering
---

# Ideal band pass and low pass filters (DT)

Signals and Systems · Chapter 6: Discrete-Time Systems

- Asked in: 2083 Bhadra (new course), 4 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/ideal-bandpass-lowpass-dt
- Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-6

## Question

Explain the impulse response of ideal band pass and low pass filters for discrete time signal.

## Short answer

Ideal lowpass filter with cutoff ω_c has h_LP[n]=sin(ω_c n)/(π n), h[0]=ω_c/π. Ideal bandpass with passband ω_1 to ω_2 has h_BP[n]=[sin(ω_2 n)-sin(ω_1 n)]/(π n) = 2 cos(ω_0 n) sin(W n)/(π n). Both are non-causal and not absolutely summable, so unrealizable.

## Answer

Ideal filters have unit gain in the passband and zero gain in the stopband. A DT frequency response is periodic with period $2\pi$, so it is defined over $-\pi\le\omega\le\pi$. The impulse response is found by the inverse DTFT, $h[n]=\frac{1}{2\pi}\int_{-\pi}^{\pi}H(e^{j\omega})e^{j\omega n}d\omega$.

#### Ideal low pass filter
$$
H_{LP}(e^{j\omega})=\begin{cases}1,&|\omega|\le\omega_c\\0,&\omega_c<|\omega|\le\pi\end{cases}
$$

$$
h_{LP}[n]=\frac{1}{2\pi}\int_{-\omega_c}^{\omega_c}e^{j\omega n}d\omega=\frac{\sin(\omega_c n)}{\pi n},\qquad h_{LP}[0]=\frac{\omega_c}{\pi}
$$

#### Ideal band pass filter
Passband $\omega_1\le|\omega|\le\omega_2$:

```text
           H(e^jw)
   +--+              +--+
   |  |              |  |  1
 --+--+------+-------+--+---> w
 -w2 -w1     0      w1  w2
```

It equals an LPF with cut-off $\omega_2$ minus an LPF with cut-off $\omega_1$:

$$
h_{BP}[n]=\frac{\sin(\omega_2 n)-\sin(\omega_1 n)}{\pi n},\qquad h_{BP}[0]=\frac{\omega_2-\omega_1}{\pi}
$$

Using $\sin A-\sin B=2\cos\frac{A+B}{2}\sin\frac{A-B}{2}$, with centre $\omega_0=\frac{\omega_1+\omega_2}{2}$ and half-width $W=\frac{\omega_2-\omega_1}{2}$:

$$
h_{BP}[n]=2\cos(\omega_0 n)\cdot\frac{\sin(Wn)}{\pi n}
$$

So the band pass impulse response is a low pass sinc (cut-off $W$) **modulated** by $2\cos\omega_0 n$, which shifts the passband to $\pm\omega_0$.

#### Remarks
- Both responses are sinc-shaped, non-zero for $n<0$: the ideal filters are **non-causal**.
- They decay only as $1/n$, so they are not absolutely summable: **not BIBO stable**, hence not realizable exactly. Practical filters truncate (window) and delay $h[n]$.

## Check yourself

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

1. How is the impulse response of an ideal filter obtained from its frequency response?
   A. Differentiate H(e^(jω)) with respect to ω
   B. Multiply H(e^(jω)) by e^(-jωn)
   C. Inverse DTFT: h[n]=(1/2π) integral of H e^(jωn) over -π to π
   D. Take the z-transform at z=0
   Answer: C. The inverse DTFT integrates over one period. (Key point: Method)

2. What is h[n] of the ideal low pass filter with cut-off ωc?
   A. sin(π n)/(ωc n)
   B. sin(ωc n)/(π n), with h[0]=ωc/π
   C. cos(ωc n)/(π n)
   D. ωc delta[n]
   Answer: B. Integrating e^(jωn) over -ωc to ωc gives the sinc form. (Key point: Low pass result)

3. How is the ideal band pass response built?
   A. LPF of cut-off ω1 plus LPF of cut-off ω2
   B. Product of two LPFs
   C. LPF of cut-off ω1 minus LPF of cut-off ω2
   D. LPF of cut-off ω2 minus LPF of cut-off ω1
   Answer: D. The difference leaves only ω1<=|ω|<=ω2. (Key point: Band pass construction)

4. Why are ideal filters not realizable?
   A. h[n] is complex valued
   B. h[n] is non-zero for n<0 (non-causal) and decays as 1/n so is not absolutely summable
   C. h[n] is zero for all n
   D. They have poles outside the unit circle that are unavoidable
   Answer: B. Non-causal and not BIBO stable, so practical filters window and delay h[n]. (Key point: Realizability)

## Related in Signals and Systems

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- [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×
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