Fall20 ECE210 L4 prelab PDF

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ECE210 Laboratory PRE-LAB#4 Fall 2020

Lab 4: Fourier transform and AM radio In Lab 4, you will finally connect all of your receiver components and tune an AM radio broadcast. You will follow the radio signal through the entire system, from antenna to loudspeaker, in the time domain and the frequency domain.

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Prelab

You should prepare for this lab by reviewing Sections 8.3 and 8.4 in the textbook on AM detection and superheterodyne receivers, familiarizing yourself with your own receiver design shown in Figure 6 in this booklet, and answering the prelab questions. Suppose you want to tune your AM receiver in the lab to an AM station broadcasting close to Champaign-Urbana ωc with a carrier frequency of fc = 2π kHz, but your receiver is prepared to decode an AM signal with carrier frequency of 14 kHz. Therefore, a previous step is needed to bring down the broadcasted signal from fc to the intermediate IF = 14 kHz. That step is achieved by multiplying (mixing) the signal at fc with a co-sinusoidal frequency fIF = ω2π LO . In our superheterodyne AM receiver, the LO signal generated by a LO (local oscillator) with frequency fLO = ω2π will be the function generator. There are two possible LO frequencies that will bring the signal from it’s broadcasted carrier frequency fc to the intermediate frequency fIF . One is the case depicted in Figure 1, where fLO1 = fC − fIF , and the other one is the case depicted in Figure 2, where fLO2 = fC + fIF .

F (ω)

MIXER

F (ω)

X(ω)

cos(ωLO) = cos(2π fLO)

AM Station

f(

fC

fIS

−fIS

−fC

Image Station

ω ) 2π

Local Oscillator(LO) X(ω) Modulation Property: 1 X(ω) = (F (ω − ωLO) + F (ω + ωLO)) 2 −fIF

−fHF

fIF

f(

fHF

ω ) 2π

Figure 1 – Diagram showing the modulation property in a graphical way for one of the two possible L.O frequencies ( fLO1 ). In this case fLO1 = fC − fIF . The “image station” problem is also shown, where another AM station located at fIS can interfere at fIF after being shifted in frequency by fLO1 .

AM Station

Image Station

F (ω) f( −fIS

fC

fC

fIS

ω ) 2π

X(ω)

−fHF

−fIF

fIF

fHF

f(

ω ) 2π

Figure 2 – Diagram showing the modulation property in a graphical way for one of the two possible L.O frequencies ( fLO2 ). In this case fLO2 = fC + fIF . The “image station” problem is also shown, where another AM station located at fIS can interfere at fIF after being shifted in frequency by fLO2 .

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1. Find the L.O. frequency (fLO1 ) needed to have the following AM stations shifted in frequency from their carrier frequencies to the intermediate frequency fIF = 14 kHz using the case shown in Figure 1, where fLO1 = fC − fIF . Also find the “image station” frequency (fIS ) and the high frequency (fHF ) where the AM station is also shifted in frequency. AM Station:

WDWS AM, fC = 1400 kHz

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WILL AM, fC = 580 kHz

fLO1 = fIS = fHF =

2. Find the L.O. frequency needed to have the following AM stations shifted in frequency from their carrier frequencies to the intermediate frequency fIF = 14 kHz using the case shown in Figure 2, where fLO2 = fC + fIF . Also find the “image station” frequency and the high frequency (fHF ) where the AM station is also shifted. AM Station:

WDWS AM, fC = 1400 kHz

WILL AM, fC = 580 kHz

fLO2 = fIS = fHF =

3. Sketch the amplitude response curve |HIF (ω )| of an ideal IF filter designed for an IF of fIF =

ωIF 2π

=14 kHz if the filter bandwidth must be 10 kHz. Label the axes of your plot carefully using appropriate tick marks. Indicate on your graph the upper and lower cut-of f frequencies in kHz. Don’t forget the negative frequencies axis.

|HIF(ω )| ω kHz 2π

0 1 τ

rect( τt

) ( use tables). Then, assuming τ = 500 µs, sketch |F (ω )| in dB scale, (i.e. sketch 20 log 10 |F (ω )|, where log 10 is the base 10 logarithm). Find all the local extrema (min. or max.) of |F (ω )|dB in the region shown below [0 - 6 kHz], fill in the table and plot. (Useful information: |sinc(x)| = 0, for |x| = nπ, where n ≥ 1 is an integer. |sinc(x)| has local maxima, computed numerically, at |x| ≈ 0, 4.493, 7.725, 10.904, 14.0662... )

4. Write down the Fourier transform of f (t) =

f (kHz) 0.00 kHz

F (ω) =

|F (f )|dB = 20 log10 (|F (f )|) 20 log10 (1) = 0 dB 20 log10 (0) = −∞

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