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-π≤ω≤π.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 forn≥L.Draw a “rough” sketch(general shape is adequate) of what you would expectˆX(ω) would look like for-π≤ω≤π.(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·L•N=L•N= 2·L
Solution:

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.

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)ejωon⇔H(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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