程序代写案例-301L

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Engineering 301L - Digital Signal Processing and Applications
Midterm Exam, 10/21/2020

Name________________________________
1. Analog to Digital (A/D) Conversion: [20 pts]
a) What is an anti-aliasing filter and why it is necessary?

b) Two important parameters that affect the quality of A/D conversion are the sampling rate
and the quantization level. If the target signal has a dynamic range of [-5V, +5V] and an 8-
bit A/D board is used, what will be the maximum quantization error (QE)?

c) Assume the measurement noise in step b) comes solely from QE. If the 8-bit A/D board
is replaced by a 12-bit A/D board, explain the expected improvement on QE and SNR?

d) In Lab-2, interpolation was utilized to estimate the other sample locations. Please
describe the change of mean-squared error (MSE) associated with varying number of data
samples and number of bits used in storage and recovery.
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2. Complex signals, signal transformation [20 pts]
a) Perform complex variable multiplication through polar representation. Show the result in
Cartesian format. [5 pts]

= 4 ∗ 2; = 1 + √3; =
1
2

1
2


b) Determine the real part of a complex variable [5 pts]
2 −
6
√3
2 +
6
√3


c) Let () = 0.5[() − ( − 4)] + ( − 4) + 2 ( − 5)
Show the plot of x(n), x(-n-2), and x(n+2) [10 pts]

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3. Determine and justify if the system is i) linear and ii) time-invariant [15 pts]

a) [()] = ∑ ()2
+0
=1 (n > 1)

b) [()] = () ()

c) [()] = () + ; a and b are real numbers
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4. Let h(n) be the impulse response of an LTI system. Prove the following conditions: [30 pts]

a) output y(n) is the linear convolution of the input x(n) and h(n)
() = () ∗ ℎ() = ∑ ()ℎ( − )

=−∞


b) linear convolution is commutative, i.e.,

() ∗ () = ∑ ()( − ) =

=−∞
∑ ()( − ) =

=−∞
() ∗ ()


c) [()()] ∗ [()ℎ()] = ()[() ∗ ℎ()] when () =







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5. Determine output of an LTI system y(n) with given inputs and impulse responses: [25 pts]

a) () = (
1
2
)

(); ℎ() = [() − ( − 9)] [10 pts]


b) () = {
1, 0 ≤ ≤ 4
0, ℎ
, ℎ() = {
, 0 ≤ ≤ 6
0, ℎ
[15 pts]

Hint: you need to determine several intervals of n for the output






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