This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Digital Signal Processing I Final Exam Fall 2010 ECE538 13 Dec.. 2010 Cover Sheet Test Duration: 120 minutes. Open Book but Closed Notes. Calculators NOT allowed. This test contains FOUR problems. All work should be done in the blue book provided. Do not return this test sheet, just return your blue book. 1 Problem 1. (a) Let x [ n ] and y [ n ] be realvalued sequences both of which are evensymmetric: x [ n ] = x [ n ] and y [ n ] = y [ n ]. Under these conditions, prove that r xy [ ℓ ] = r yx [ ℓ ] for all ℓ . (b) Express the autocorrelation sequence r zz [ ℓ ] for the complexvalued signal z [ n ] = x [ n ]+ jy [ n ] where x [ n ] and y [ n ] are realvalued sequences, in terms of r xx [ ℓ ], r xy [ ℓ ], r yx [ ℓ ], and r yy [ ℓ ]. (c) Determine a closedform expression for the autocorrelation sequence r xx [ ℓ ] for the signal x [ n ] below. x [ n ] = braceleftBigg sin( π 2 n ) πn bracerightBigg (1) (d) Determine a closedform expression for the autocorrelation sequence r yy [ ℓ ] for the signal y [ n ] below. y [ n ] = ( 1) n x [ n ] = ( 1) n braceleftBigg sin( π 2 n ) πn bracerightBigg (2) (e) Determine a closedform expression for the autocorrelation sequence r zz [ ℓ ] for the complexvalued signal z [ n ] formed with x [ n ] and y [ n ] defined above as the real and imaginary parts, respectively, as defined below. You must show all work and simplify as much as possible. z [ n ] = x [ n ] + jy [ n ] (3) (f) Plot r zz [ ℓ ]. 2 Problem 2. For all parts of this problem, the reconstructed spectrum is computed according to the equation below: Y r ( ω ) = N 1 summationdisplay k =0 Y N ( k ) sin bracketleftBig N 2 parenleftBig ω 2 πk N parenrightBigbracketrightBig N sin bracketleftBig 1 2 parenleftBig ω 2 πk N parenrightBigbracketrightBig e j N...
View
Full
Document
This note was uploaded on 02/19/2012 for the course ECE 538 taught by Professor Zoltolski during the Fall '08 term at Purdue University.
 Fall '08
 ZOLTOLSKI
 Digital Signal Processing, Signal Processing

Click to edit the document details