Homework_8_Solutions

Homework_8_Solutions - Homework 8 Solutions 8.1-2: (b)...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Homework 8 Solutions 8.1-2: (b) 0.01ݏ݅݊ܿ ሺ100ߨݐሻ ֞ 0.0001∆ሺ ସ଴଴గ The bandwidth of this signal is 200π rad/s or 100 Hz. Therefore the Nyquist rate is 200 Hz (samples/sec). (c) ݏ݅݊ܿሺ100ߨݐሻ + 3ݏ݅݊ܿ ሺ60ߨݐሻ ֞ 0.01ݎ݁ܿݐ ቀ ଶ଴଴గ ቁ + ଶ଴ ∆ሺ ସ଴଴గ The bandwidth of ݎ݁ܿݐሺ ଶ଴଴గ is 50 Hz and that of ∆ሺ ଶସ଴గ is 60 Hz. The bandwidth of the sum is the higher of the two, that is, 60 Hz. The Nyquist sampling rate is 120 Hz. 8.1-7 (a) ܺሺݓሻ = ∆ቀ ଶ଴గ ቁ + ߨ[ߜሺݓ + 20ߨሻ + ߜሺݓ − 20ߨሻ] The bandwidth is 10 Hz. There is an impulse at 10 Hz, as seen from X(w). The Nyquist rate is 20 Hz. Hence, f s = 10 Hz will not permit reconstruction of x(t). This is verified from the sampled signal spectrum, shown as a function of f in Hz. (b) The Nyquist rate is 20 Hz. Hence the sampling rate f s = 20 Hz is adequate despite the fact that x(t) contains an impulse at the highest frequency 10 Hz. This is because the impulse component is cos(20πt). To reconstruct x(t) from the spectrum shown below, we need an ideal LPF of cutoff frequency 10 Hz and gain T = 1/20. Because the rect function value is 0.5 at the edge (cutoff), the LPF gain at the cutoff frequency 10 Hz is 0.5 × ଶ଴ = ସ଴ . Hence for the input of an impulse of strength 40π at ±10 Hz, the output will be an impulse of strength 50π at ݂ = ±10 Hz. Hence, the filter output is the spectrum: ܺሺݓሻ = ∆ቀ ଶ଴గ ቁ + ߨ[ߜሺݓ + 20ߨሻ + ߜሺݓ − 20ߨሻ] and
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
ݔሺݐሻ = 5ݏ݅݊ܿ ሺ5ߨݐሻ + cos ሺ20ߨݐሻ (c) In part (b), we found the cutoff B
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/14/2011 for the course BME 343 taught by Professor Emelianov during the Spring '09 term at University of Texas.

Page1 / 5

Homework_8_Solutions - Homework 8 Solutions 8.1-2: (b)...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online