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# Digital Transmission over Passband Channels

Module by: Roy Ha, Mohammad Borran, Dinesh Rajan. E-mail the authors

Summary: (Blank Abstract)

For several reasons, including multiplexing many streams of data on a single broadband channel, and FCC (Federal Communications Commission) regulations, in most of the practical cases, the digitally modulated baseband signal has to be translated in frequency domain to have nonzero energy only in a designated band. Simple cosines can help translate well designed signals in baseband to passband, around the frequency of interest. This center frequency is usually referred to as carrier frequency, and the cosine signal used for the translation is known as the carrier signal.

## Carrier Amplitude Modulation (Amplitude Shift Keying, ASK)

Multiplying the baseband PAM signal waveforms by a carrier signal, shifts their spectrum by the carrier frequency and, thus, places the signal into the passband of the channel. This is called amplitude modulation.

m,m12M: u m t= A m g T tcos2π f c t m m 1 2 M u m t A m g T t 2 f c t
(1)
Recall that the Fourier transform of the carrier is δf f c +δf+ f c 2 δ f f c δ f f c 2 . Since multiplication in the time domain corresponds to the convolution in the frequency domain, the spectrum of the amplitude-modulated signal in Equation 1 is
U m f= A m 2( G T f f c + G T f+ f c ) U m f A m 2 G T f f c G T f f c
(2)
.

When the transmitted pulse shape is rectangular, the amplitude-modulated carrier signal is usually called amplitude-shift keying (ASK). Note that impressing the baseband signals onto the amplitude of the carrier signal does not change the basic geometric representation of the digital PAM signal waveforms. The signal space spanned by the signal set is still of dimension one.

ψ 1 t= s 1 t E s =2 E g g T tcos2π f c t ψ 1 t s 1 t E s 2 E g g T t 2 f c t
(3)

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