e j 2 \u03c0 2 n 3 7 17 12 ej 2\u03c0 2n 37 e j 2 \u03c0 2 n 37 e j2 \u03c0 3 n 3 7 e j2 \u03c03 n 37 1

E j 2 π 2 n 3 7 17 12 ej 2π 2n 37 e j 2 π 2 n 37 e

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ECE 513 Exam #2 Instructor: Dr. Cranos Williams 6 4.Spectral Estimation: (20 Pts)We are given the following continuous-time analog signalx1(t) = cos(2·π·F1t).(14)whereF1= 190Hz(a) Given that the sampling frequency isFs= 500Hz, draw the magnitude of thetrue spectrum,X(ω), of the sampled signalx(n),-∞ ≤n≤ ∞for-πωπ.Be sure to clearly label all relevant frequencies,ω, on the x-axis.(b) A rectangular window of sizeL= 40 is used to make the sequence finite suchthat the new windowed sequence ˆx(n) = 0 fornL.Draw a “rough” sketch(general shape is adequate) of what you would expectˆX(ω) would look like for-πωπ.(c) We want to take anNpoint DFT of the signal ˆx(n) to produceˆX(k). Circle allconditions onNthat will allow us to recreate ˆx(n), 0< n < L-1 fromˆX(k)N= 0.5·LN=LN= 2·L Solution:
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ECE 513 Exam #2 Instructor: Dr. Cranos Williams 7 𝜋 0.76𝜋 െ0.76𝜋 𝜔 𝑋ሺ𝜔ሻ Figure 2: Magnitude response of ˆ X ( ω ) (b) A rough sketch of the magnitude of ˆ X ( ω ) is given in Fig. 2 (c) Since N should be greater than or equal to L , N = L and N = 2 L are the correct answers.
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ECE 513 Exam #2 Instructor: Dr. Cranos Williams 8 5.FIR Filter Design: (20 Pts)The magnitude response of an ideal low pass filter is shown in Figure 3.Hd(°)°c°1Figure 3: Ideal Lowpass Filter Response.and the impulse response for this ideal lowpass filter is given ash(n) =(sin(ωcn)πnn6= 0ωcπn= 0(18)The passband and stopband criteria for the filter that we want to design isPassband: 0 to 1500 HzStopband: 2500 to 4000 HzFs = 8000 HzUSEFUL DTFT PROPERTIES:h(n)eonH(ω-ωo),Frequency Shifting Propertyh(n-n0)H(ω)e-jωn0,Time Shifting Property(a) What is an appropriate value for the digital frequencyωcifωcis considered to bethe center of the transition band?
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