Analog Sinusoidal Modulation

1. Analog Sinusoidal Modulation

2. Analog communication Transmit/receive analog waveforms Amplitude Modulation (AM) Freq. Modulation (FM) Phase Modulation (PM) Quadrature Amplitude Mod. Pulse Amplitude Modulation Digital communication Same but treat transmission and reception as digitized Amplitude Shift Keying (ASK) Freq. Shift Keying (FSK) Phase Shift Keying (PSK) QAM PAM SignalProcessing CarrierCircuits Transmission Medium Carrier Circuits SignalProcessing s(t) r(t) TRANSMITTER CHANNEL RECEIVER Single-Carrier Modulation Methods

3. Radio Frequency (RF) Modem • Message signal: stream of bits • Digital sinusoidal modulation in digital signaling • Analog sinusoidal modulation in carrier circuits for upconversion to RF Error Correction Digital Signaling D/A Converter SignalProcessing CarrierCircuits Transmission Medium Carrier Circuits SignalProcessing s(t) r(t) TRANSMITTER CHANNEL RECEIVER

4. Modulation • 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 A(t) is real-valued amplitude function fc is carrier frequency (t) is real-valued phase function

5. Review Amplitude Modulation by Cosine • Multiplication in time: convolution in Fourier domain (let w0 = 2 pf0): • Sifting property of Dirac delta functional • Fourier transform property for modulation by a cosine

6. lower sidebands Y(w) ½F(w + w0) ½F(w - w0) F(w) ½ 1 w -w0 - w1 -w0 + w1 w0 - w1 w0 + w1 0 -w0 w0 w -w1 w1 0 Review Amplitude Modulation by Cosine • Example:y(t) = f(t) cos(w0 t) Assume f(t) is an ideal lowpass signal with bandwidth w1 Assume w1 << w0 Y(w) is real-valued if F(w) is real-valued • Demodulation: modulation then lowpass filtering • Similar derivation for modulation with sin(w0 t)

7. Review Amplitude Modulation by Sine • Multiplication in time is convolution in Fourier domain • Sifting property of the Dirac delta functional • Fourier transform property for modulation by a sine

8. Y(w) j ½F(w + w0) -j ½F(w - w0) F(w) j ½ 1 w0 w0 - w1 w0 + w1 w -w0 - w1 -w0 + w1 -w0 w -j ½ -w1 w1 0 Review Amplitude Modulation by Sine • Example: y(t) = f(t) sin(w0 t) Assume f(t) is an ideal lowpass signal with bandwidth w1 Assume w1 << w0 Y(w) is imaginary-valued if F(w) is real-valued • Demodulation: modulation then lowpass filtering lower sidebands

9. Amplitude Modulated (AM) Radio • Double sideband large carrier (DSC-LC) Carrier wave varied about mean value linearly with baseband message signal m(t) ka is the amplitude sensitivity, ka > 0 Modulation factor is  = kaAm 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)

10. Rf Rs C Rl + vs(t) – Amplitude Modulation • 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 simpleenvelope detector • Diode (with forwardresistance Rf ) in series • Parallel connection ofcapacitor C and loadresistor Rl

11. Amplitude Modulation (con’t) • Let Rsbe 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 Rlbetween 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

12. Other Amplitude Modulation Types • Double sideband suppressed carrier (DSB-SC) • Double sideband variable carrier (DSB-VC) • Single sideband (SSB): Remove either lower sideband or upper sideband by • Extremely sharp bandpass or highpass filter, or • Phase shifters using a Hilbert transformer

13. Quadrature Amplitude Modulation • Allows DSB-SC signals to occupy same channel bandwidth provided that the two message signals are from independent sources • 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)

14. SignalProcessing CarrierCircuits Transmission Medium Carrier Circuits SignalProcessing s(t) r(t) TRANSMITTER CHANNEL RECEIVER 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

15. Frequency Modulation • Non-linear, time-varying, has memory, non-causal • For single tone message m(t) = Am cos(2 pfmt) • Modulation index is  = f / fm  << 1 => Narrowband FM (looks like double-sideband AM)  >> 1 => Broadband FM Instantaneous frequency

16. Carson's Rule • Bandwidth of FM for single-tone message at fm • Narrowband: • Wideband: • Carson’s rule for single-tone FM: • For a general message signal, fm= W

17. Summary • General form of modulation: h(t) is the impulse response of a bandpass filter or phase shifter toeffect a cancellation of one pair of redundant sidebands.

18. Optional Angle Modulation • Angle modulation general form Ac is constant carrier amplitude i(t) is instantaneous angle of modulation in radians • Average frequency inrad/s over interval Dt • Instantaneousfrequency in rad/s • Instantaneous angle Phase modulation Frequency modulation