EE445S Real-Time Digital Signal Processing Laboratory

Handout I: Analog Sinusoidal Modulation Summary

Many ways exist to modulate a message signal $m(t)$ to produce a modulated (transmitted) signal $x(t)$. For amplitude, frequency, and phase modulation, modulated signals can be expressed in the same form as

x(t) = A(t) \cos( 2 \pi f_c t + \Theta(t) )

where $A(t)$ is a real-valued amplitude function (a.k.a. the envelope), $f_c$ is the carrier frequency, and $\Theta(t)$ is the real-valued phase function. Using this framework, several common modulation schemes are described below. In the table below, the amplitude modulation methods are double sideband larger carrier (DSB-LC), DSB suppressed carrier (DSB-SC), DSB variable carrier (DSB-VC), and single sideband (SSB). The hybrid amplitude-frequency modulation is quadrature amplitude modulation (QAM). The angle modulation methods are phase and frequency modulation.
Modulation $A(t)$ $\Theta(t)$ Carrier Type Use
DSB-LC $A_c \left[ 1 + k_a m(t) \right]$ $\Theta_0$ Yes Amplitude AM radio
DSB-SC $A_c m(t)$ $\Theta_0$ No Amplitude  
DSB-VC $A_c m(t) + \epsilon$ $\Theta_0$ Yes Amplitude  
SSB $A_c \sqrt{m^2(t) + \left[ m(t) \star h(t) \right]^2}$ $\arctan( - { {m(t) \star h(t)} \over {m(t)} } )$ No Amplitude $\dagger$ Marine radios
QAM $A_c \sqrt{m_1^2(t) + m_2^2(t)}$ $\arctan( - { {m_2(t)} \over {m_1(t)} } )$ No Hybrid Satellite
Phase $A_c$ $\Theta_0 + k_p   m(t)$ No Angle Underwater
Frequency $A_c$ $2 \pi k_f \int_0^t   m(t)   dt$ No Angle FM radio
          TV audio
$\dagger$ $h(t)$ is the impulse response of a bandpass filter or phase shifter to effect a cancellation of one pair of redundant sidebands. For ideal filters and phase shifters, the modulation is amplitude modulation because the phase would not carry any information about $m(t)$.

Each analog TV channel is allocated a bandwidth of 6 MHz. The picture intensity and color information are transmitted using vestigal sideband modulation. Vestigal sideband modulation is a variant of amplitude modulation (not shown above) in which the upper sideband is kept and a fraction of the lower sideband is kept, or vice-versa. In an analog TV signal, the audio portion is frequency modulated.

The following quantity is known as the complex envelope

\tilde{x}(t) = A(t) \; e^{ j   \Theta(t) } = x_{I}(t) + j   x_{Q}(t)

where $x_{I}(t)$ is called the in-phase component and $x_{Q}(t)$ is called the quadrature component. Both $x_{I}(t)$ and $x_{Q}(t)$ are lowpass signals, and hence, the complex envelope $\tilde{x}(t)$ is a lowpass signal. An alterative representation for the modulated signal $x(t)$ is

x(t) = \Re{e} \{ \tilde{x}(t)   e^{j 2 \pi f_c t} \}

Brian L. Evans