114_1_sol_midterm_2008_Part1

# 114_1_sol_midterm_2008_Part1 - UCLA Dept. of Electrical...

This preview shows pages 1–4. Sign up to view the full content.

UCLA Dept. of Electrical Engineering EE 114, Winter 2008 Midterm Exam Solutions 1. ( a ) Let x ( n ) = [ a ;b;c;d ]. The 4-point DFT is determined as: X ( k ) = N ¡ 1 X n =0 x ( n ) e ¡ j 2 …kn=N (1) X (0) = N ¡ 1 X n =0 x ( n ) = 0 : 5( a + b + c + d ) (2) X (1) = N ¡ 1 X n =0 x ( n ) e ¡ j…k= 2 = 0 : 5 / ( a ¡ jb ¡ c + jd ) (3) X (2) = N ¡ 1 X n =0 x ( n ) e ¡ j…k = 0 : 5 / ( a ¡ b + c ¡ d ) (4) X (3) = N ¡ 1 X n =0 x ( n ) e ¡ j… 3 k= 4 = 0 : 5 / ( a + jb ¡ c ¡ jd ) (5) (i) If x ( n ) is assumed even and periodic with period 4, then x ( n ) = x ( N ¡ n ). This implies that b = d . This leads to: X (0) = ( a + 2 b + c ) (6) X (1) = ( a ¡ c ) (7) X (2) = ( a ¡ 2 b + c ) (8) X (3) = ( a ¡ c ) (9) (ii) If x ( n ) is assumed odd and periodic with period 4,then x ( n ) = ¡ x ( N ¡ n ). This implies b = ¡ d , c = ¡ c , and a = ¡ a . This leads to c = 0 and a = 0, and further: X (0) = 0 (10) X (1) = ¡ j 2 b (11) X (2) = 0 (12) X (3) = j 2 b (13) 1

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

View Full Document
2. ( a ) Let x ( n ) = [3 ; ¡ 2 ; 1 ; 4] and y ( n ) = [4 ; 4 ; 2 ; 2 ; 1 ; 1]. x ( n ) / y ( n ) = [12 ; 4 ; 2 ; 22 ; 17 ; 11 ; 7 ; 5 ; 4] (14) ( b ) Let x ( n ) = [2 ; ¡ 2 ; 1 ; ¡ 1] and y ( n ) = [ ¡ 4 ; 4 ; ¡ 2 ; 2 ; ¡ 1 ; 1]. C xy ( k ) = [2 ; ¡ 4 ; 7 ; ¡ 10 ; 15 ; ¡ 20 ; 14 ; ¡ 8 ; 4] : (15) 2
3. The given tube model can be decoupled into three separate sources of resonance: the back cavity, the front cavity, and the Helmholtz resonance due to the coupling of the two.

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

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

## This note was uploaded on 11/12/2011 for the course EE 114 taught by Professor Vanschaar during the Spring '11 term at UCLA.

### Page1 / 5

114_1_sol_midterm_2008_Part1 - UCLA Dept. of Electrical...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online