6.003 Homework #12 Solutions
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1 6.003 Homework # Solutios Problems. Which are rue? For each of the D sigals x [] through x 4 [] below), determie whether the coditios listed i the followig table are satisfied, ad aswer for true or F for false. x [] x [] x 3 [] x 4 [] Xe j0 ) = 0 Xe j ) d = 0 Xe j ) is purely imagiary e jk Xe j ) is purely real for some iteger k F F F F F F F F x [] x [] x 3 [] x 4 []= x 4 [ + 5]
2 6.003 Homework # Solutios / Fall 0. Iverse Fourier he magitude ad agle of the Fourier trasform of x[] are show below. magitude si 0 agle / / Determie x[]. x[] ) Xe j ) = j si e j/ e j/ e j/ = j e j/ = j e j x[] = Xe j )e j d = e j ) e j d = e j ) e j) d 4 4 / = 0 = / = 0 otherwise
3 6.003 Homework # Solutios / Fall 0 3 Egieerig Desig Problem 3. Samplig with alteratig impulses A C sigal x c t) is coverted to a D sigal x d [] as follows: { xc ) eve x d [] = x c ) odd a. Assume that the Fourier trasform of x c t) is X c j) show below. X c j) Determie the D Fourier trasform X d e j ) of x d []. Let x e [] = ) x d [] = x c ). he x e [] = X e e j )e j d = ) x d [] = ) X d e j )e j d = e j X d e j )e j d = X d e j )e j ) d = X d e j +) )e j d hus X e e j ) = X d e j+) ). Also x e [] = X e e j )e j d = x c ) = W X c j)e j d Substitutig = we see that X ee j ) = X cj) = >. It follows that provided X c j) = 0 for X d e j ) = X cj) = as show i the followig figure. X d e j ) W +W A alterative solutio is to costruct a ovel samplig fuctio that cosists of alteratig impulses hece the title of the problem). p a t) = ) δt ) = You ca thik of this sigal as two times a impulse trai with period mius a impulse trai with period p a t) = δt ) δt ) = =
4 6.003 Homework # Solutios / Fall 0 4 he Fourier trasform is the P a j) = δ = = = odd which is show below 3 δ ) ) P a j) = δ ) Covolvig this fuctio of frequecy with X c j) ad covertig to the DF leads to the same solutio for X d e j ) that was show above. b. Assume that x c t) is badlimited to W W. Determie the maximum value of W for which the origial sigal x c t) ca be recostructed from the samples x d []. From the previous part, W must be greater tha 0. hus W <. 3
5 6.003 Homework # Solutios / Fall Boxcar samplig A digital camera focuses light from the eviromet oto a imagig chip that coverts the icidet image ito a discrete represetatio composed of pixels. Each pixel represets the total light collected from a regio of space x d [, m] = md+ md D+ D x c x, y) dx dy where is a large fractio of the distace D betwee pixels. his kid of samplig is ofte called boxcar samplig to distiquish it from the ideal impulse samplig that we described i lecture. Assume that boxcar samplig is defied i oe dimesio as x d [] = + x c t) dt where is the itersample time. a. Let X c j) represet the cotiuous-time Fourier trasform of x c t). Determie the discrete-time Fourier trasform X d e j ) of x d [] i terms of X c j),, ad. If we defie pt) = { < t < 0 otherwise the we ca express x d [] as x b ) where x b t) = x c p)t). he Fourier trasform of x b t) is the the product of the Fourier trasforms of x c t) ad pt). P j) = e jt dt = si Provided that X b j) is 0 for >, X d e j ) = X bj) = = X cj)p j)) = = si X c j ) over the iterval < < ad repeats periodically with period ) outside that iterval. b. Assume that x c t) is badlimited to W W. Determie the the maximum value of W for which the origial sigal x c t) ca be recostructed from the samples x d []. Compare your aswer to the aswer for a ideal impulse sampler. Sice X b j) must be 0 for > see above), it follows that X cj) must be zero over this same rage. herefore, X c must be badlimited i W =. Furthermore, P j) is o-zero for < /, so all of the iformatio i X c ca be recostructed from that i X b i.e., there is o loss of iformatio i goig from X c to X b ). hus boxcar samplig has exactly the same frequecy limitatios as impulse samplig.
6 6.003 Homework # Solutios / Fall 0 6 c. Describe the effect of boxcar samplig o the resultig samples x d []. How are the samples that result from boxcar samplig differet from those that result from impulse samplig? he samples geerated by boxcar samplig are lowpass filtered relative to those geerated by impulse samplig. he atteuatio is greatest ) at the maximum frequecy = ad for the largest possible value of, which is. he atteuatio is less for lower frequecies ad for smaller values of.
7 6.003 Homework # Solutios / Fall D processig of C sigals Samplig ad recostructio allow us to process C sigals usig digital electroics as show i the followig figure. x c t) impulse sampler x d [] y d [] impulse y p t) h d [] recostructio ideal LPF y c t) he impulse sampler ad impulse recostuctio use samplig iterval = /00. he uit-sample fuctio h d [] represets the uit-sample respose of a ideal D lowpass filter with gai of for frequecies i the rage < <. he ideal LPF passes frequecies i the rage 00 < < 00. It also has a gai of throughout its pass bad. Assume that the Fourier trasform of the iput x c t) is Xj) show below. X c j) Determie Y c j) Impulse samplig of x c t) produces x d [] = x c ). Substitutig ito the trasform relatios shows that x d [] = X d e j )e j d = x c ) = X c j)e j d hus X d e j ) = X cj) = sice d = d. herefore X d e j ) = X c j ) = 00 X c ) j. /00 Notice that the cutoff frequecy = 00 maps to = = =. Also, X de j ) is periodic i as are all D Fourier trasforms, as show below. X d e j ) 00 he output of the ideal lowpass filter has the followig trasform. Y d e j ) 00 Impulse recostructio the geerates a sigal with the followig trasform.
8 6.003 Homework # Solutios / Fall 0 8 Y p j) he fial output has the followig trasform, where the DC value is 00 =. Y c j) he overall effect is that the iput sigal is lowpass filtered by the discrete system.
6.003 Homework #12 Solutions
6.003 Homework # Solutios Problems. Which are rue? For each of the D sigals x [] through x 4 [] (below), determie whether the coditios listed i the followig table are satisfied, ad aswer for true or F
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