Reference no: EM132193490

Digital signal processing

Question 1. We have a Discrete-Time (DT) sinusoidal LTI (Linear Time-Invariant) system that is described by the difference equation:

y(n) = ∑_k=0^M b_kx(n - k) - ∑_k=0^N a_ky(n-k) 

here: b - {1,2,1}, a - {2,1,2}, x(n) - {0,1,2,2,1,0}.
- Find the sequence (show the counting intermediate steps) y(n);
- Sketch graphically the given DT LTI the system.

Question 2. We have a discrete time signal x(n) - {0, 1, 2, 2, 1, 0}. Find a normalized autocorrelation sequence for this signal (show the calculation of the intermediate steps) ρ(l) and sketch it graphically.

r_xx(l) = Σ_n=i^N-|k|-1 x(n)x(n-k),

here: i=l, k=0 when l≥ 0, and i=0, k=l when l≤ 0.

Question 3. z-transformation

Determine the z-transform of the signal x(n) - {0, 1, 2, 2, 1, 0} and sketch it graphically.

Write H(z) characteristic from task 1 (DT LTI system).

Note: please see "Frequency response of LTI Systems"

DT LTI system has zeros z1 = -1 and z2 = -1, poles are p1= -0.25+0.97i and p2= - 0.25-0.97i. Draw a zeros-poles diagram for this system. Is this system stable?

Question 4. The given DT signal is s(n) = 2sin(0.25πn)+ sin(0.5πn). Show the amplitude of this signal spectrum with DFT (discrete Fourier transform) has 16 points.

Question 5. The given DT signal is x(n) - {1, 0, 1, 0, 1, 0, 1, 0}. Locate signal's DFT using Radix-2 Fast Fourier transform algorithm (FFT) algorithm. When DFT ({1, 1}) - {2, 0}, DFT ({0, 0}) - {0, 0}.

Book chapters to look: "Digital Signal Processing" John G Proakis, Dimitris G. Manolakis
1. 2.4 Discrete-Time Systems Described by Difference Equations
2.5 Implementation of Discrete-Time Systems
2. 2.6 Correlation of Discrete-Time Signals
3. 3.1 The z-Transform
Properties of the z-Transform
Rational z-Transforms
3.5 Analysis of Linear Time-Invariant Systems in the z-Domain

Frequency-Domain Characteristics of Linear TIme-Invariant Systems
Frequency Response of LTI Systems
4. 4.2 Frequency Analysis of Discrete-Time Signals
Frequency-Domain and Time-Domain Signal Properties
Properties of the Fourier Transform for Discrete-Time Signals

5. 8.1 Efficient Computation of the OFT: FFT Algorithms