EE-410L Communication Systems Lab · Lab 8 · OPEN ENDED: Frequency Modulation and Demodulation (Simulation)All labs

The message moves the frequency. The amplitude never changes.

Lab 8 is open ended: you write the FM simulation yourself, in an m-file and in Simulink. Read this page first, move the sliders, and use the plan at the end.

Frequency modulation

FM signals(t) = cos(2πf_ct + 2πk_f∫m(τ)dτ)the integral of the message is the phase

Instantaneous frequencyf_i(t) = f_c + k_fm(t)k_f in Hz per volt

Deviation and indexΔf = k_f|m|_max, β = Δf/f_mfor a tone message

In FM the carrier frequency follows the message: a positive message voltage pushes it up, a negative voltage pulls it down, and the amplitude stays the same. The sensitivity k_f is the frequency change for one volt, and the largest change is the frequency deviation Δf. A voltage controlled oscillator (VCO) does exactly this: its input voltage sets its output frequency.

Keep Δf below f_c. If the deviation is larger than the carrier, the instantaneous frequency would go negative and the signal no longer looks like FM. Also sample at least ten times faster than the highest frequency, f_c + Δf.

The quadrature modulator

Phaseθ(t) = 2πk_f∫m(τ)dτthe integrator of figure 1

I and QI = cos θ, Q = sin θtwo baseband signals

OutputI cos 2πf_ct − Q sin 2πf_ct= cos(2πf_ct + θ)

Figure 1 of the manual first integrates the message to get the phase θ(t). Figure 2 then shifts the carrier phase by θ with a quadrature (IQ) modulator: cos θ multiplies the carrier, sin θ multiplies the carrier shifted by 90°, and the two products are subtracted. By the identity cos(a + b) = cos a cos b − sin a sin b the result is cos(2πf_ct + θ), the FM signal. So a phase modulator fed with the integral of the message is a frequency modulator. In MATLAB the integral is cumsum(m)/fs; in Simulink it is an Integrator block, or a Continuous-Time VCO block does the whole job.

Spectrum and bandwidth

Linesf_c ± n·f_man endless set, spaced f_m apart

Heights|J_n(β)|Bessel functions of the first kind

Carson's ruleB ≈ 2(Δf + f_m) = 2f_m(β + 1)about 98 % of the power

A tone FM signal has lines at the carrier and at every multiple of f_m on both sides. A small β (narrowband FM) gives one significant pair, like AM. A large β gives many pairs and a wide spectrum. The carrier line itself is J₀(β): it shrinks as β grows and disappears at β ≈ 2.405, while the total power stays the same. Changing the carrier frequency only moves the whole spectrum; changing k_f changes Δf and β, and so the width.

Demodulation

Differentiatords/dt = −2πf_i(t)·sin(…)an AM signal, v_o = −RC dv_i/dt

Envelope2π(f_c + k_fm(t))rectify and low pass

Messagek_fm(t)remove the constant f_c

Section 8.3 recovers the message by frequency discrimination. Differentiating cos φ(t) gives −φ'(t) sin φ(t): the amplitude of the result is φ'(t) = 2πf_i(t), which follows the message. That is an AM signal, so an envelope detector (rectifier and low pass filter) gets the envelope back, and removing its constant part leaves k_fm(t). A constant input gives no output from a differentiator, which is why only the frequency, not the steady carrier, reaches the detector.

Why the small ripple? The demo uses a moving average over one carrier period as its low pass filter. A sharper filter, or a carrier much faster than the message, gives a cleaner result. In MATLAB, fmdemod or a PLL does the same job.

Your open ended work

m-file

Message m(t)→cumsum to the phase→cos(2πf_c t + θ)→diff→abs and low pass→remove DC

plot the message, the FM signal, its spectrum and the recovered message in one figure with subplot

Simulink

Sine Wave→Integrator and Gain 2πk_f→IQ modulator or VCO→Derivative→Abs and filter→Scope

use a fixed step solver with a step much shorter than one carrier period

The objectives of the manual ask you to show three effects, so plan one figure for each: (1) the same message with two or three carrier frequencies, (2) the same carrier with two or three values of k_f (and so of Δf and β), and (3) the spectrum of each case with its Carson bandwidth marked. Then demodulate one case and compare the recovered message with the original. Write down f_s, f_c, f_m, k_f, Δf and β for every figure.

A starting point for the m-file. fs = 400e3; t = 0:1/fs:5e-3; fm = 500; fc = 20e3; kf = 8e3; m = cos(2*pi*fm*t); theta = 2*pi*kf*cumsum(m)/fs; s = cos(2*pi*fc*t + theta); d = [0 diff(s)]*fs; then rectify d, filter it, and remove its mean.

Check yourself

Q1What is frequency modulation?

A form of analog modulation in which the instantaneous frequency of the carrier changes in proportion to the message voltage, fi(t) = fc + kf·m(t), while the amplitude stays constant.

Q2Why is the message integrated before the modulator?

Frequency is the rate of change of phase. To make the frequency follow m(t), the phase must follow the integral of m(t). The integrated message is then applied as a phase shift, which a quadrature (IQ) modulator can do.

Q3What are the frequency deviation and the modulation index?

The deviation Δf = kf·|m|max is the largest shift of the carrier frequency away from fc. For a tone of frequency fm the modulation index is β = Δf/fm; it decides how many sidebands are significant.

Q4How much bandwidth does an FM signal need?

In theory infinite, because the lines at fc ± n·fm never stop. In practice Carson's rule gives B ≈ 2(Δf + fm), which holds about 98 % of the power. Broadcast FM with Δf = 75 kHz and 15 kHz audio needs about 180 kHz.

Q5How does the differentiator demodulator work?

The derivative of cos φ(t) is −φ'(t)·sin φ(t), so its amplitude is proportional to the instantaneous frequency. The output is an AM signal whose envelope is fc + kf·m(t); an envelope detector recovers it and removing the DC leaves the message.

Q6What happens to the FM signal when the carrier frequency is changed, and when kf is changed?

Changing fc moves the whole spectrum up or down; the deviation, the index and the bandwidth stay the same. Changing kf changes Δf and β: the carrier swings further, more sidebands appear and the bandwidth grows.