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EE104: Lecture 22 Outline Announcements HW due, final HW posted (due Monday, 3/17) Final exam Thursday, 3/20, 8:30-11:30am More details next week Review of Last Lecture Spectral Analysis of FM FM Bandwidth and Carson’s Rule Narrowband FM Modulation Wideband FM Modulation FM Detection

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Review of Last Lecture Vestigial Sideband Modulation Transmits a small part of unwanted sideband Does not cause distortion at baseband Can use a carrier or suppress the carrier Standard envelope detection or DSBSC demod. AM Radio Superheterodyne Receivers Use IF downconversion and processing Gets around reradiation and processing at f c. Introduction to FM

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Information signal encoded in carrier frequency (or phase): s(t)=A c cos( (t)) (t)=f(m(t)) Standard FM: (t)=2 f c t+2 k f m( )d Signal robust to amplitude variations Robust to signal reflections and refractions Instantaneous frequency: f i =f c +k f m(t) Maximum Deviation: f=f c +k f |m(t)| Bandwidth of S(f) depends on f.

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Spectral Analysis of FM s(t)=A c cos(2 f c t+2 k f m( )d Nonlinear function of m(t) Very hard to analyze for general m(t). Let m(t)=cos(2 f m t): Bandwidth f m Spectrum S(f) is a sequence of delta functions at multiples of f m from f c fcfc f c +f m f c +2f m f c +3f m f c + 4f m f c -4f m f c -3f m f c -2f m f c -f m …… f S(f) for m(t)=cos(2 f m t) NBFM B 2B m WBFM B2fB2f

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FM Bandwidth and Carson’s Rule Frequency Deviation: f=k f max|m(t)| Maximum deviation of f i from f c : f i =f c +k f m(t) Carson’s Rule: B depends on maximum deviation from f c AND how fast f i changes Narrowband FM: f<<B m B 2B m Wideband FM: f>>B m B 2 f B 2 f+2B m

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Generating FM Signals NBFM WBFM Direct Method: Modulate a VCO with m(t) Indirect Method: Use a NBFM modulator, followed by a nonlinear device and BPF m(t) Product Modulator A c sin(2 f c t) s(t) 2 k f (·)dt (t) -90 o LO + A c cos(2 f c t) + -

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FM Detection Differentiator and Envelope Detector Zero Crossing Detector Uses rate of zero crossings to estimate f i Phase Lock Loop (PLL) Uses VCO and feedback to extract m(t)

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Main Points Spectral analysis of FM difficult. For a simple cosine information signal, FM spectrum is discrete and infinite. FM signal bandwidth depends on information signal amplitude. Carson’s rule yields B=2B m +2 f. NBFM easy to analyze and is generated with a simple product modulator. WBFM harder to generate and to analyze. In theory just need a differentiator and envelope detector for FM demodulation. Multiple methods used in practice

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