Lesson 29 – Modulation 2

Lesson 29 – Modulation 2#

Learning Outcomes#

  1. Analyze AM modulators in the frequency domain for both sinusoidal and non-sinusoidal inputs.

  2. Understand and determine the bandwidth of an AM signal.

  3. Understand block diagram Frequency Domain Multiplexing (FDM) systems.

  4. Analyze FDM systems in the frequency domain.

Modulation 2#

AM with More Than One Frequency#

Although it is instructive to consider how to use a single frequency as a message signal, it is not very practical. In fact, most systems want to send information containing a bandwidth of frequencies. Any communications system, for example, wants to ensure it can at least use data in the audible range (20 Hz – 22 kHz) as its message signal.

Everything up to this point still applies to a message signal with more than one frequency. For example, consider the following modulation system:

Where \(v_m(t)\) is now:

\[ v_m(t) = 2\cos(360^\circ \cdot 1k \cdot t) + \cos(360^\circ \cdot 3k \cdot t) + 6\cos(360^\circ \cdot 5k \cdot t)\ \text{V} \]

In this case, determining the time domain representation of the output signal (\(v_{AM}\)) is overly complicated. Even using the distributive law to write the equation of the output signal is overly tedious. However, it is perfectly feasible to draw this signal in the frequency domain if we recall that each frequency in the message signal produces two frequencies in the output signal:

  • One at \(f_c + f_m\)

  • One at \(f_c - f_m\)

The amplitude of each frequency is:

\[ \frac{A_m A_c}{2} \]

So, the example above produces six frequencies in the output.

First, notice how the spectrum is symmetric around the carrier frequency, \(f_c\).

The frequencies to the right of the carrier are:

  • \(590k + 1k = 591k\ \text{Hz}\)

  • \(590k + 3k = 593k\ \text{Hz}\)

  • \(590k + 5k = 595k\ \text{Hz}\)

The amplitudes are:

\[ A_{m1} = \frac{2 \cdot 2}{2} = 2\ \text{V} \]
\[ A_{m2} = \frac{1 \cdot 2}{2} = 1\ \text{V} \]
\[ A_{m3} = \frac{6 \cdot 2}{2} = 6\ \text{V} \]

The frequencies to the left are mirror images about \(f_c\) with identical amplitudes.

AM Spectrum Construction Steps#

  1. Add the carrier frequency to each message frequency.

  2. Compute amplitudes using \(\frac{A_m A_c}{2}\).

  3. Reflect the results about the carrier frequency.

Bandwidth#

Bandwidth is the range of frequencies used to transmit a signal.

\[ BW = f_{high} - f_{low} \]

For this signal:

\[ BW = 595k - 585k = 10k\ \text{Hz} \]

The bandwidth of an AM signal is always:

\[ BW = 2f_{m,\text{max}} \]

Digital Modulation#

The carrier must be sinusoidal, but the message can be analog or digital.

Amplitude Shift Keying (ASK)#

  • Binary 0 → low amplitude (0 V)

  • Binary 1 → high amplitude (e.g., 5 V)

Frequency Shift Keying (FSK)#

  • Binary 0 → one frequency

  • Binary 1 → another frequency

Frequency Division Multiplexing (FDM)#

Two approaches to multiplexing:

  • Time Division Multiplexing (TDM): Signals share time

  • Frequency Division Multiplexing (FDM): Signals share frequency spectrum

FDM allows multiple signals to be transmitted simultaneously by assigning each a unique frequency band.

FDM Process#

  1. Band-limit each signal (using LPFs)

  2. Shift each signal to a unique frequency band (modulation)

  3. Add signals together

The result is a composite signal with non-overlapping spectra.

Low-pass filters ensure signals do not overlap by limiting bandwidth.

Example Problem 1#

Two music signals are multiplexed as shown below. Graph the output in the frequency domain.

Understand#

We are multiplexing signals in the frequency domain.

Note: \(f_c\) represents:

  • Filter cutoff frequency (LPFs)

  • Carrier frequency (modulators)

Identify#

  • Knowns: Input frequencies, LPF cutoff, carrier frequencies

  • Unknowns: Output spectrum

  • Assumption: System supports full bandwidth

Plan#

  • Follow each signal path

  • Apply filtering

  • Apply modulation

  • Combine results

Solve#

First signal after LPF:

Second signal after LPF:

Then:

  • Shift first signal by \(120k\)

  • Reflect about carrier

  • Shift second signal by \(155k\)

  • Reflect about carrier

LPFs ensure no spectral overlap.

Answer#

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