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Unformatted text preview: ECE 2200: Section V Problems (Week 6 / Spring 2009)
1. The FIR ﬁlter with impulse response h[n] = 2δ [n − 2] has output y [n] = u[n − 3] − u[n − 6] Determine the input x[n]. 2. The FIR ﬁlter has a step starting at sample zero as its input, i.e. x[n] = u[n], and the output is the Kronecker delta function, i.e. y [n] = δ [n]. Determine the impulse response of the ﬁlter. 3. A linear time-invariant system is described by the diﬀerence equation y [n] = x[n] − 2x[n − 1] + x[n − 2]
ˆ (a) Find the frequency response H (ej ω ), and then express it as a mathematical formula, in polar form (magnitude and phase). ˆ (b) Plot the magnitude and phase of H (ej ω ) as a function of ω for −π ≤ ω ≤ π . ˆ ˆ Do this by hand and with the MATLAB function freqz. (c) Find all frequencies, ω , for which the response to the input ejωn is zero. (d) When the input to the system is x[n] = sin(πn/100)determine the functional form for the output signal y [n]. (e) Impulse Response: Determine the response of this system to a unit impulse input. Plot h[n] as a function of n . 4. For each of the following frequency responses determine the corresponding impulse response :
ˆ ˆ (a) H (ej ω ) = 1 + 2e−3j ω ˆ ˆ (b) H (ej ω ) = 2e−3j ω cos(ˆ ) ω ω ˆ ˆ sin(5ˆ ) (c) H (ej ω ) = e−4.5j ω sin(ˆ /2) ω 1 5. The frequency response of a linear time-invariant lter is given by the formula
ˆ ˆ ˆ ˆ H (ej ω ) = (1 + e−j ω )(1 − e−jπ/3 e−j ω )(1 − ejπ/3 e−j ω ) (a) Write the diﬀerence equation that gives the relation between the input x[n] and the output y [n]. (b) What is the impulse response of this system?
ˆ (c) If the input is of the form x[n] = Aejφ ej ωn , for what values of −π ≤ ω ≤ π ˆ will y [n] = 0 for all n? 6. A linear time-invariant system is dened by the frequency response plots (magnitude and phase) given in the ﬁgure. Use the graphs to determine the output of the system y [n] to the input x[n] = 10 + 10 cos(0.5πn) 2 ...
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This note was uploaded on 09/30/2009 for the course ECE 2200 taught by Professor Johnson during the Spring '05 term at Cornell.
- Spring '05