1 Theory - IIT Hyderabad

EE5300: Digital Signal Processing, Fall 2014
Indian Institute of Technology Hyderabad
HW 1, Assigned: Wednesday 20.08.2014.
Theory Due: Friday 29.08.2014 in class, Code Due: Friday 29.08.2014 by 11:59 pm.
Submission instructions
• Solutions to problems must be turned in at the beginning of class.
• Code must be uploaded to the Google Drive folder that will be shared. Create a subfolder (name = your roll number)
and place your code in it.
1
Theory
1. The continuous time signal xc (t) = sin(20πt)+cos(40πt) is sampled with a sampling period T to obtain the discrete-time
signal x[n] = sin( π5 ) + cos( 2π
5 ).
(a) Find a choice for T consistent with this information. (2)
(b) Is your answer unique? If yes, why? If not, find another choice for T . (3)
2. Find x[n] = xc (nT ) for the following:
• xc (t) = cos(2π(1000)t), T = (1/3000) sec. (1)
• xc (t) = sin(2π(1000)t), T = (1/1500) sec. (1)
• xc (t) = sin(2π(1000)t)/πt, T = (1/5000) sec. (3)
3. We discussed in class the discrete time processing of analog signals with specific attention to discrete-time LTI systems.
Assume that the discrete time LTI system is an ideal low pass filter with cutoff ωc , the sampling period is T , the
highest frequency of the signal is ΩN . Derive the relation between these quantities so that no aliasing is present in the
continuous time output. Also assume that the reconstruction filter is an ideal LPF with cutoff frequency π/T . (5)
4. A signal xc (t) is sampled at rate T1 to get x[n] and reconstructed at rate T2 to get yc (t). ΩN = 2π × 5 × 103 and
Xc (jΩ) is triangular with Xc (j0) = 1 and Xc (jΩ) becoming zero at Ω = ±ΩN . x[n] is acted upon by an ideal low pass
filter H(ejω ) with ωc = π2 and unity gain. Sketch and label Yc (jΩ) for the following cases: (i) T1 = T2 = 10−4 , (ii)
T1 = 0.5 × 10−4 , T2 = 10−4 , (iii) T1 = 10−4 , T2 = 0.5 × 10−4 . (5)
5. Consider the following:
• Xc (jΩ) = 0, |Ω| ≥ 2π × 104 ,
• x[n] = xc (nT ),
n
P
• y[n] =
x[k].
k=−∞
(a) Find the largest sampling period T to avoid aliasing (to exactly recover xc (t) from x[n]). (1)
(b) Find h[n]. (2)
(c) In terms of X(ejω ), what is the value of y[n] for n = ∞? (2)
6. A continuous time signal xc (t) is processed in the discrete domain with an LTI system defined by H(ejω ) = jω/T, π ≤
ω ≤ π, T = 1/10 sec. The resulting signal is converted to yc (t).
(a) Find yc (t) for the following xc (t).
i. xc (t) = cos(6πt). (1)
ii. xc (t) = cos(14πt). (1)
(b) Are the outputs those you would expect from a differentiator? (3)
7. A continuous time signal xc (t) is sampled with a sampling period of T1 (with Xc (jΩ) = 0, |Ω| ≥ π/T1 ) that results in
x[n]. x[n] is reconstructed at a sampling rate T2 to give yc (t). For the general case T1 6= T2 , express yc (t) in terms of
xc (t). Is the relationship different for T1 > T2 and T1 < T2 ? (5)
1
8. Consider a system that samples a continuous time signal and processes it with a discrete LTI system. Assume that
Xc (jΩ) = 0, |Ω| ≥ 4000π. Find
( the largest value of the sampling period T and the corresponding frequency response
|Ω|Xc (jΩ), 1000π ≤ |Ω| ≤ 2000π,
(5)
H(ejω ) such that Yc (jΩ) =
0, otherwise.
9. Downsampling: Given that x[n] = xc (nT ) and y[n] = x[2n]. Further, y[n] is reconstructed to yc (t) at a sampling period
of T 0 . Answer the following:
(a) Assume that Xc (jΩ) = 0, |Ω| > 2π(100). What value of T is required so that X(ejω ) = 0, π2 < |ω| ≤ π? (3)
(b) How should T 0 be chosen so that yc (t) = xc (t)? (2)
10. Which of the following signals can be downsampled by a factor of 2 without loss of information?
(a) x[n] = δ[n − n0 ]. (1)
(b) x[n] = cos(πn/4). (1)
(c) x[n] = cos(πn/4) + cos(3πn/4). (1)
(d) x[n] = sin(πn/3)/(πn/3). (1)
(e) x[n] = (−1)n sin(πn/3)/(πn/3). (1)
2
Programming – Fun with Sampling
In this problem, you will see the effect of sampling and aliasing. You can work with wav files from the source http:
//aacapps.com/lamp/voices.
1. Read a female voice file and a male voice file using wavread and play it back. You can use the sound function to do the
playback. In this problem playback the files at the sampling rate returned by the wavread function. (5)
2. Now repeat the playback at the following sampling rates: 40,000, 35,000, 25,000, and 8,000 . Note down your observations. Can you explain your observations? You can rewrite the files at these sampling rates using wavwrite. This
experiment should help you understand how voices in the movie Star Wars may have been generated – especially Darth