Page 1 of 5 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. Page 2 of 5 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] Page 3 of 5 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 Page 4 of 5 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 () = Page 5 of 5 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 欢迎咨询51作业君
