---
title: "Output for sinusoid mix input, h=(1/2)^n u[n]"
subject: "Digital Signal Analysis and Processing"
chapter: 3
key: "semester_7/digital_signal_analysis_and_processing:lti-output-sinusoid-mix"
source: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/lti-output-sinusoid-mix
site: BCT Engineering
---

# Output for sinusoid mix input, h=(1/2)^n u[n]

Digital Signal Analysis and Processing · Chapter 3: Analysis of LTI System in Frequency Domain

- Asked in: 2073 Shrawan, 7 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/digital_signal_analysis_and_processing/lti-output-sinusoid-mix
- Whole chapter: https://www.bctengineer.com/md/semester_7/digital_signal_analysis_and_processing/chapter-3

## Question

Determine the output sequence of the system with impulse response h[n] = (1/2)^n u[n] when the input signal is x[n] = 10 − 5 sin(πn/2) + 20 cos(πn), −∞ < n < ∞.

## Short answer

Frequency response H(e^{jω}) = 1/(1 - 0.5 e^{-jω}). At ω=0: H=2; ω=π/2: H=0.8944∠-26.57°; ω=π: H=2/3. DC output 20, -5 sin becomes -4.472 sin(πn/2 - 26.57°), 20 cos becomes 13.33 cos(πn). Output: y[n]=20 - 4.472 sin(πn/2 - 26.57°) + 13.33 cos(πn).

## Answer

h[n] = (½)ⁿ u[n] has frequency response

H(e^(jω)) = 1/(1 − ½e^(−jω))

The input is a sum of sinusoids, and for each sinusoid A cos(ω₀n + φ) the output is A|H(ω₀)| cos(ω₀n + φ + ∠H(ω₀)). By linearity, add the outputs.

x[n] = 10 − 5 sin(πn/2) + 20 cos(πn): three components, at ω = 0, π/2 and π.

| Component | ω₀ | H(e^(jω₀)) | \|H\| | ∠H |
|---|---|---|---|---|
| 10 (DC) | 0 | 1/(1 − ½) = 2 | 2 | 0 |
| −5 sin(πn/2) | π/2 | 1/(1 + 0.5j) = 0.8 − 0.4j | 0.8944 | −26.57° (−0.4636 rad) |
| 20 cos(πn) | π | 1/(1 + ½) = 2/3 | 0.6667 | 0 |

(At ω = π/2: e^(−jπ/2) = −j, so H = 1/(1 − ½(−j)) = 1/(1 + 0.5j).)

**Outputs:**
- DC: 10 × 2 = 20
- Sine: −5 × 0.8944 × sin(πn/2 − 0.4636) = −4.472 sin(πn/2 − 0.4636)
- cos(πn): 20 × (2/3) cos(πn) = 13.33 cos(πn) = 13.33(−1)ⁿ

**Answer:**

y[n] = 20 − 4.472 sin(πn/2 − 26.57°) + 13.33 cos(πn),  −∞ < n < ∞

(Since the input is defined for all n, this steady-state result is the complete output.)

## Check yourself

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

1. How is the output found for an input made of several sinusoids?
   A. Convolve the sum with h[n] only at w = 0
   B. Apply H at the highest frequency to all terms
   C. Scale and phase-shift each sinusoid by H at its frequency, then add the results
   D. Multiply the inputs together
   Answer: C. LTI linearity lets each component be processed separately by its H(e^(jw0)). (Key point: Superposition)

2. At which frequencies are the three components of the input?
   A. 0, pi/4 and pi/2
   B. pi/2, pi and 2pi
   C. 0, pi and 2pi
   D. 0, pi/2 and pi
   Answer: D. 10 is DC (w = 0), sin(pi n/2) has w = pi/2 and cos(pi n) has w = pi. (Key point: Frequencies)

3. What is the DC term of the output?
   A. 20
   B. 10
   C. 5
   D. 13.33
   Answer: A. H(e^(j0)) = 1/(1 - 0.5) = 2, so 10 x 2 = 20. (Key point: DC gain)

4. What are |H| and angle of H at w = pi/2?
   A. About 2 and 0
   B. About 0.894 and -26.57 degrees
   C. About 0.667 and 0
   D. About 0.894 and +26.57 degrees
   Answer: B. H = 1/(1 + 0.5j) = 0.8 - 0.4j, magnitude 0.894, phase -26.57 degrees. (Key point: Gain at pi/2)

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