{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Homework_8_Solutions

# Homework_8_Solutions - Homework 8 Solutions 8.1-2(b 0.01J...

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

Homework 8 Solutions 8.1-2: (b) 0.01ݏ݅݊ܿ ሺ100ߨݐሻ֞ 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 ሺ60ߨݐሻ֞0.01ݎ݁ܿݐቀ 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

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

View Full Document
ݔሺݐሻ= 5ݏ݅݊ܿ
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern