2.15: Amplitude Modulation
- Page ID
- 126962
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Learning Objectives: Amplitude Modulation
- Explain what amplitude modulation is and how it works in time and frequency
- Explain basic demodulation in frequency
- Understand how signal bandwidth determines the spacing of carrier frequencies in AM systems
YouTube Videos
Amplitude Modulation
Demodulating AM Signals
Summary
Amplitude Modulation (AM)
\[ y(t)=x(t)\cos(\omega_c t) \]
\(\cos(\omega_c t)\): carrier signal with a higher frequency
\(x(t)\): modulating signal containing information
How high should the modulation frequency be?
\[ y(t)=x(t)\cdot\cos(\omega_c t) \;\;\leftrightarrow\;\; Y(j\omega)=\frac{1}{2\pi}\left[X(j\omega)*\left(\pi\delta(\omega-\omega_c)+\pi\delta(\omega+\omega_c)\right)\right] \]
\[ Y(j\omega)=\frac{1}{2}\,X\!\big(j(\omega-\omega_c)\big)+\frac{1}{2}\,X\!\big(j(\omega+\omega_c)\big) \]
\(\omega_c\) large enough so the 2 replicas of \(X(j\omega)\) do not overlap.
Figure 2.15.1: Spectrum of the output signal when x(t) is multiplied by a carrier signal \(\cos(9\omega_M)\)
Figure 2.15.1 shows that a whole copy of the object spectrum is displaced at a higher frequency, i.e., \(\omega_c = \pm 9\,\omega_M\)
We need a demodulated system to bring back the replicas @ 0 frequency axis
Frequency Division Multiplexing (FDM)
Definition: Frequency Division Multiplexing (FDM) is a technique for transmitting multiple signals simultaneously by assigning each signal a unique carrier frequency.
In theory, we can retrieve (i.e., demodulate) each signal
Figure 2.15.2: Output spectrum when a complex signal is multiplied by a carrier signal \(\cos(9\omega_M)\). The individual components of the combined signal can be transmitted simultaneously without overlap.
Figure 2.15.3: Visual representation of AM V.S. FM.
\[ y(t)=x(t)\cos(\omega_c t) \]
AM = Amplitude Modulation → \(x(t)\) is modulated with a cosine with high frequency, modulating its amplitude.
FM = Frequency Modulation → next class
Demodulation
How can we recover the original signal \(x(t)\) from the modulated signal \(y(t)\)?
Step 1 to recover the signal is
Figure 2.15.4: Flow diagram of the first step to demodulation.
\[ r(t)=y(t)\cos(\omega_c t) \]
\[ R(j\omega)=\frac{1}{2}\,Y\!\big(j(\omega-\omega_c)\big)+\frac{1}{2}\,Y\!\big(j(\omega+\omega_c)\big) \] scaled version of the input spectrum
Figure 2.15.5: Representation of step 1 to demodulation being applied to the signal from Figure 2.15.1.
Step 2: Apply lowpass filter
\[ H_D(j\omega)= \begin{cases} 1, & |\omega|\le \omega_M\\ 0, & |\omega|>\omega_M \end{cases} \]
\(y(t)\;\xrightarrow{\ \times\cos(\omega_c t)\ }\;r(t)\;\xrightarrow{\ H_D(j\omega)\ }\;s(t)\)
\[ S(j\omega)=R(j\omega)\cdot H_D(j\omega) \]
The demodulated cosine should have exactly the same frequency and phase as the carrier signal.
Figure 2.15.6: Representation of step 2 to demodulation being applied to the signals from Figure 2.15.1 & 2.15.5.
\(S'(j\omega)\) has the same spectrum as the original spectrum \(X(j\omega)\) but with a different gain, i.e., \[ S'(j\omega)=\frac{1}{2}X(j\omega) \] → by changing the volume (i.e., gain) then we can match \[ X(j\omega)=2S'(j\omega) \]