6.003 Homework 12
Due at the beginning of recitation on Wednesday, May 5, 2010. Problems
1. Sampling CT sinusoids
Consider 3 CT signals:
x 1(t ) = cos(3000t ) ,
x 2(t ) = cos(4000t ) , and
x 3(t ) = cos(5000t ) .
Each of the is sampled as follows
x 1[n ]= x 1(nT ) ,
x 2[n ]= x 2(nT ) , and
翠竹黄花
x 3[n ]= x 3(nT ) ,
where T =0.001. Which of the resulting DT signals has the highest DT frequency? Which has the lowest DT frequency?
2. Sampling with alternating impuls
A CT signal x c (t ) is converted to a DT signal x d [n ] as follows: x c
(nT ) n even x d [n ]= −x c (nT ) n odd
a. Assume that the Fourier transform of x c (t ) is X c (jω) shown below.
ω
X c (jω)
W
1
Determine the DT Fourier transform X d (e j Ω) of x d [n ].
b. Assume that x c (t ) is bandlimited to −W ≤ ω ≤ W . Determine the maximum value of W for which the original signal x c (t ) can be reconstructed from the samples x d [n ].不吝赐教的意思
c. Make a diagram of a system to reconstruct x c (t ) from x d [n ].
2
6.003 Homework 12 / Spring 2010
息泽神君3. Boxcar sampling A digital camera focus light from the environment onto an imaging chip that converts the incident image into a discrete reprentation compod of pixels. Each pixel reprents the total light collected from a region of space
mD +∆2 nD +∆2 x d [n,m ]= x c (x,y ) dx dy
mD −∆2nD −∆2
where ∆ is a large fraction of the distance D between pixels. This kind of sampling is often called “boxcar” sampling to distinquish it from the ideal “impul” sampling that we described in lecture. Assume that boxcar sampling is defined in one dimension as
nT +∆2 x d [n ]= x c (t ) dt
nT −∆2儿童康复训练
大连警察学校where T is the intersample “time.”
a. Let X c (jω) reprent the continuous-time Fourier transform of x c (t ). Determine the discrete-time Fourier transform X d (e j Ω) of x d [n ] in terms of X c (jω), ∆, and T .
b. Assume that x c (t ) is bandlimited to −W ≤ ω ≤ W . Determine the the maximum value of W for which the original signal x c (t ) can be reconstructed from the samples x d [n ]. Compare your answer to the answer for an ideal “impul” sampler.
c. Describe the effect of boxcar sampling on the resulting samples x d [n ]. How are the samples that result from boxcar sampling different from tho that result from impul sampling?
4. DT processing of CT signals
shown in the following figure.
x c (t )y c (t )
莫测高深The “impul sampler” and “impul reconstuction” u sampling interval T = π/100.The unit-sample function h d [n ] reprents the unit-sample respon of an ideal DT low-Sampling and reconstruction allow us to process CT signals using digital electronics as pass filter with gain of 1 for frequencies in the range −π < Ω < π The “ideal LPF” 2 2 . pass frequencies in the range −100 <ω< 100. It also has a gain of T throughout its pass band.
Assume that the Fourier transform of the input x c (t ) is X (jω) shown below.
ω
X c (jω)
−100100
半时
1
Determine Y c (jω).
M IT OpenCourWare
ocw.mit.edu
6.003 Signals and Systems
Spring 2010
For information about citing the materials or our Terms of U, visit: ocw.mit.edu/terms.名茶小镇
