--- title: "Band pass and band limited signals; LTI properties" subject: "Communication Systems" chapter: 1 key: "semester_6/communication_systems:bandpass-bandlimited-lti-properties" source: https://www.bctengineer.com/q/semester_6/communication_systems/bandpass-bandlimited-lti-properties site: BCT Engineering --- # Band pass and band limited signals; LTI properties Communication Systems · Chapter 1: Introduction - Asked in: 2068 Bhadra (CS I), 4+4 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/bandpass-bandlimited-lti-properties - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1 ## Question Define Band pass signal and Band Limited signal with example. Write the properties of LTI system. ## Short answer Band-limited signal has spectrum zero beyond W Hz (e.g., telephone speech 3.4 kHz). Band-pass signal has spectrum concentrated around carrier fc >> W (e.g., AM at 1 MHz). LTI system properties: linearity, time-invariance, described by impulse response convolution, frequency response Y(f)=H(f)X(f), commutative/associative/distributive convolution, causality if h(t)=0 for t<0, BIBO stability if ∫|h(t)|dt<∞. ## Answer #### Band-limited signal A signal $g(t)$ is **band-limited** to $W$ Hz if its spectrum is zero beyond $W$: $$ G(f)=0\quad\text{for}\ \lvert f\rvert>W $$ Its energy lies between $0$ and $W$ (it is a low-pass or baseband signal). **Examples:** telephone speech limited to 3.4 kHz; $g(t)=\text{sinc}(2Wt)$, whose spectrum is a rectangle from $-W$ to $W$; audio after a 20 kHz anti-aliasing filter. Such signals can be sampled without loss at $f_s\ge 2W$. #### Band-pass signal A **band-pass signal** has its spectrum concentrated in a band of width $2W$ around a carrier frequency $f_c$, with $f_c\gg W$, and is zero near $f=0$: $$ G(f)\ne0\ \ \text{only for}\ f_c-W\le\lvert f\rvert\le f_c+W $$ It can be written as $g(t)=g_I(t)\cos2\pi f_ct-g_Q(t)\sin2\pi f_ct$. **Examples:** an AM broadcast signal at 1 MHz with 10 kHz bandwidth; an FM station at 100 MHz; $m(t)\cos2\pi f_ct$ (DSB-SC). ```text Band-limited (low-pass) Band-pass |G(f)| |G(f)| +--+--+ +--+ +--+ | | | | | | | --+--+--+---> f ------+--+----+----+--+---> f -W 0 W -fc 0 fc ``` #### Properties of an LTI system 1. **Linearity (superposition):** $a x_1+b x_2\to a y_1+b y_2$; includes homogeneity and additivity. 2. **Time invariance:** $x(t-t_0)\to y(t-t_0)$. 3. **Complete description by impulse response:** $y(t)=x(t)*h(t)$. 4. **Frequency response:** $Y(f)=H(f)X(f)$; a sinusoid in gives a sinusoid of the **same frequency** out, with changed amplitude and phase. No new frequencies are created. 5. **Commutative, associative and distributive** with respect to convolution: cascaded LTI systems give $h_1*h_2$; parallel systems give $h_1+h_2$. 6. **Causality:** causal if $h(t)=0$ for $t<0$. 7. **Stability (BIBO):** stable if $\int_{-\infty}^{\infty}\lvert h(t)\rvert dt<\infty$. 8. **Power spectra:** for random inputs $S_y(f)=\lvert H(f)\rvert^2S_x(f)$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. A signal is band-limited to W Hz if A. Its spectrum is zero for |f| > W B. Its time duration is limited to W seconds C. Its spectrum is nonzero only around fc D. Its amplitude never exceeds W Answer: A. Band-limited means G(f) = 0 beyond W, so all energy lies between 0 and W. (Key point: Band-limited definition) 2. Which is a band-pass signal? A. sinc(2Wt) B. Telephone speech limited to 3.4 kHz C. An FM station at 100 MHz D. A DC level Answer: C. A band-pass signal has its spectrum concentrated around a carrier fc >> W and is zero near f = 0. (Key point: Band-pass signal) 3. A sinusoid is applied to an LTI system. The output is A. A sinusoid of the same frequency with changed amplitude and phase B. A sinusoid with harmonics added C. A sinusoid of double frequency D. Always zero Answer: A. LTI systems create no new frequencies; they only scale amplitude and shift phase via H(f). (Key point: Frequency response of LTI) 4. For an LTI system, which pair of conditions is correct? A. Causal: h(t)=0 for t>0; stable: ∫h dt = 0 B. Causal: h(t)=0 for t<0; stable: ∫|h(t)|dt finite C. Causal: h(t) even; stable: h(t) bounded D. Causal: ∫|h|dt finite; stable: h(t)=0 for t<0 Answer: B. Causality needs h(t) = 0 for t < 0, and BIBO stability needs absolute integrability of h(t). 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