# 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.

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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

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

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.

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 B2fB2f

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

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) + -

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)

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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