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EE445S Real-Time Digital Signal Processing Lab Fall 2011 Analog Sinusoidal Modulation Prof. Brian L. Evans Dept. of Electrical and Computer Engineering The University of Texas at Austin Lecture 19 Outline • Introduction • Amplitude sinusoidal modulation Double-sideband large carrier Quadrature amplitude modulation Other amplitude modulation types • Frequency modulation • Conclusion 19 - 2 Example: Radio Frequency Modem • Message signal: stream of bits • Digital sinusoidal modulation in digital signaling • Analog sinusoidal modulation in carrier circuits for upconversion to radio frequencies (RF) Error Correction m[k ] Signal Processing Digital Signaling Carrier Circuits TRANSMITTER D/A Converter Transmission Medium s(t) CHANNEL Carrier Circuits r(t) Signal Processing mˆ [ k ] RECEIVER 19 - 3 Modulation • Some characteristic of a carrier signal is varied in accordance with a modulating signal • For amplitude, frequency, and phase modulation, modulated signals can be expressed as y(t ) f (t ) cos(2 f c t (t )) f(t) is real-valued amplitude function fc is carrier frequency (t) is real-valued phase function • See Modulation handout (Appendix I) 19 - 4 Amplitude Modulation • By cosine y t f t cosc t 1 Y F 2 c c • Fourier property Y 1 1 F c F c 2 2 • Spectrum F() is Shifted left by c and scaled by ½ and Shifted right by c and scaled by ½ • By sine y t f t sin c t 1 Y F 2 j c c • Fourier property Y j j F c F c 2 2 • Spectrum F() is Shifted left by c and scaled by j/2 and Shifted right by c and 19 - 5 scaled by –j/2 Amplitude Modulation by Cosine • Example: y(t) = f(t) cos(c t) Assume f(t) is an ideal lowpass signal with bandwidth 1 Assume 1 << c lower sidebands F() ½F c 1 Y() ½F c ½ -1 0 1 -c - 1 c -c + 1 0 c - 1 c c + 1 Y() is real-valued if F() is real-valued • Demodulation: modulation then lowpass filtering • Similar derivation for modulation with sin(c t) 19 - 6 Amplitude Modulation by Sine • Example: y(t) = f(t) sin(c t) Assume f(t) is an ideal lowpass signal with bandwidth 1 Assume 1 << c lower sidebands F() j ½F c 1 Y() j -1 0 1 -j ½F c ½ c - 1 -c - 1 c c c + 1 -c + 1 -j ½ Y() is imaginary-valued if F() is real-valued • Demodulation: modulation then lowpass filtering 19 - 7 Amplitude Modulated (AM) Radio • Double sideband large carrier (DSC-LC) Carrier wave varied about mean value linearly with baseband message signal m(t) y (t ) Ac 1 ka m(t ) cos( 2 f c t ) Ac cos( 2 f c t ) Ac ka m(t ) cos( 2 f c t ) ka is the amplitude sensitivity, ka > 0 Modulation factor is = ka Am where Am is maximum amplitude of m(t) • Envelope of s(t) has about same shape as m(t) if | ka m(t) | < 1 for all t fc >> W where W is bandwidth of m(t) 19 - 8 Amplitude Modulated (AM) Radio • Disadvantages Redundant bandwidth is used Carrier consumes most of the transmitted power • Advantage Simple detectors (e.g. AM radio receivers for cars) • Receiver uses a simple envelope detector Diode (with forward Rs resistance Rf ) in series Parallel connection of + capacitor C and load vs(t) – resistor Rl Rf C Rl 19 - 9 Amplitude Modulated (AM) Radio • Let Rs be source resistance • Charging time constant (Rf + Rs) C must be short when compared to 1/ fc, so (Rf +Rs) C << 1/ fc • Discharging time constant Rl C Long enough so that capacitor discharges slowly through load resistor Rl between positive peaks of carrier wave Not so long that capacitor voltage will not discharge at max rate of change of modulating wave 1/fc << Rl C << 1/W 19 - 10 Quadrature Amplitude Modulation • Allows DSB-SC signals to occupy same channel bandwidth provided that the two message signals are from independent sources y(t ) Ac m1 (t ) cos(2 f c t ) Ac m2 (t ) sin( 2 f c t ) A(t ) cos(2 f c t (t )) m2 (t ) 2 2 A(t ) Ac m1 (t ) m2 (t ) (t ) arctan m1 (t ) • Two message signals m1(t) and m2(t) are sent Ac m1(t) is in-phase component of s(t) Ac m2(t) is quadrature component of s(t) 19 - 11 Other Amplitude Modulation Types • Double sideband suppressed carrier (DSB-SC) s(t ) Ac m(t ) cos(2 f c t ) • Double sideband variable carrier (DSB-VC) s(t ) Ac m(t ) cos(2 f c t ) cos(2 f c t ) • Single sideband (SSB) removes either lower sideband or upper sideband by Extremely sharp bandpass or highpass filter, or Phase shifters using Hilbert transformer (slides 15-7 to 15-10) 19 - 12 Frequency Modulated (FM) Radio • Message signal: analog audio signal • Transmitter Signal processing: lowpass filter to reject above 15 kHz Carrier circuits: sinusoidal modulatation from baseband to FM station frequency (often in two modulation steps) • Receiver Carrier circuits: sinusoidal demodulation from FM station frequency to baseband (often in two demodulation steps) Signal processing: lowpass filter to reject above 15 kHz m(t ) Signal Processing Carrier Circuits TRANSMITTER Transmission Medium s(t) CHANNEL Carrier Circuits r(t) Signal Processing mˆ (t ) RECEIVER 19 - 13 Frequency Modulation • Non-linear, time-varying, has memory, non-causal t s (t ) Ac cos i (t ) Ac cos 2 f c t 2 k f m(t ) dt 0 • For single tone message m(t) = Am cos(2 fm t) f i (t ) 2 f c t sin( 2 f m t ) where f k f Am fm 1 d Instantaneous f i (t ) i (t ) f c f cos( 2 f m t ) frequency 2 dt • Modulation index is = f / fm << 1 => Narrowband FM (looks like double-sideband AM) >> 1 => Broadband FM 19 - 14 Carson's Rule • Bandwidth of FM for single-tone message at fm Narrowband: Wideband: BT 2 f m BT 2f • Carson’s rule for single-tone FM: BT 2 f m (1 ) FM Radio f fm Peak freq. deviation (F) Peak message freq. (W) Deviation ratio (D) Bandwidth BT = 2 fm (1+ ) Station Spacing 75 kHz 15 kHz 5 180 kHz 200 kHz TV Audio 25 kHz 15 kHz 1.66 80 kHz 6 MHz • For message signal of bandwidth W, let fm = W19 - 15 Conclusion • Amplitude modulation Digital and analog versions may be used in same system Analog amplitude modulation is one method for upconversion • Double sideband amplitude modulation Transmission bandwidth is twice message bandwidth (wasteful) • Quadrature amplitude modulation Uses cosine and sine to modulate two different message signals and subtracts resulting waveforms Two messages in same transmission bandwidth (efficient) 19 - 16