20085ee102_1_HW-6

# 20085ee102_1_HW-6 - H ( s ) = K ( s-0) ( s-1)( s + 1) 2...

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

FALL 2008: Put First Letter of LAST Name in the corner →→± ( Otherwise Your HW May Be LOST! ) PRINT: (LAST , Middle, First):——————————————————– HW: # 6 A LATE HW CANNOT BE A HW! Posted: W, November 12 Hand In To Me: M, November 24 Attach This Sheet To Your HW (Otherwise It May Be Lost!) 1. (i) Given the periodic signal x ( t ) (over one period) in Fig.1. Compute X n then write down the Fourier Sine-Cosine series of x ( t ). (ii) The periodic signal of part (i) is now applied to a system S whose system function H ( s ) admits the P-Z-P (Poles-Zeroes-Plot) shown in Fig. 2. Moreover, H (2) = 2 9 . Write down the Fourier series of the corresponding output. (iii) Compute the LSE when x ( t ) is approximated by the ﬁnite FS: x 1 ( t ) := 1 ± n = - 1 X n e in π t . Big Hints: (i) T = 2 ω 0 = 2 π T = π . X 0 = 1 2 ² 1 0 t dt = 1 4 , X n = 1 2 ² 1 0 e - in π t t dt, = ? , n ² = 0 , and n : even , = ? , n : odd . f ( t ) : real f ( t ) = F 0 + ± n =1 2 A n cos n π t + ± n =1 2( - B n ) sin( n π ) t. 1

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

View Full Document
where A n := Re X n =? , n : even , A n =? , n : odd , B n := Im X n =? , n : even , B n =? , n : odd . (ii) Look at Fig. 2 then you see that H ( s ) can be expressed as:
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: H ( s ) = K ( s-0) ( s-1)( s + 1) 2 Where K is a constant to be determined by you. Ask yourself: Why do I need to have K ? 2. (a) The amplitude and the phase of a periodic signal x ( t ) , with period T , is | X n | = 1 n 2 + 1 , Θ n = n π 2 , n = 0 , ± 1 , ± 2 , ..... Write down the Fourier Sine-Cosine series of x ( t ). (b) The signal x ( t ) of part (a) is now applied to a L, TI, C system de-scribed by d 2 y ( t ) dt 2 + 2 dy ( t ) dt + y ( t ) = x ( t ) , y (0) = 0 = ˙ y (0) Compute the amplitude and phase of the corresponding output. 3. Write down the FS representation of f ( t ) := sin 2 t cos 3 t,-∞ < t < ∞ . 4. Compute the FT F ( · ) of the following signals: (i) f ( t ) = e at , t < , = e-at , t ≥ , where a > 0. (ii) f ( t ) = te-t U ( t ) In each case calculate |F ( i ω ) | and θ ( ω ). 5. Text, #5.2. 2...
View Full Document

## This note was uploaded on 04/03/2009 for the course EE 102 taught by Professor Levan during the Fall '08 term at UCLA.

### Page1 / 3

20085ee102_1_HW-6 - H ( s ) = K ( s-0) ( s-1)( s + 1) 2...

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

View Full Document
Ask a homework question - tutors are online