EE456 Digital Communications Professor Ha Nguyen September 2016 - - PowerPoint PPT Presentation

Chapter 3: Frequency Modulation (FM)

EE456 – Digital Communications

Professor Ha Nguyen September 2016

EE456 – Digital Communications 1

Chapter 3: Frequency Modulation (FM)

Angle Modulation

In AM signals the information content of message m(t) is embedded as amplitude variation of the carrier. Two other parameters of the carrier are frequency and phase. They can also be varied in proportion to the message signal, which results in frequency-modulated and phase-modulated signals. Frequency modulation (FM) and phase modulation (PM) are closely related and collectively known as angle modulation. In our study, we will mainly focus on FM.

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Chapter 3: Frequency Modulation (FM)

Instantaneous Frequency

Consider a generalized sinusoidal signal c(t) = A cos θ(t), where θ(t) is the generalized angle and is a function of t. Over the infinitesimal duration of ∆t between [t1, t2], draw a tangential line of θ(t), which can be described by equation ωct + θ0. It is clear from the figure that, over the interval t1 < t < t2 one has: c(t) = A cos θ(t) = A cos(ωct + θ0), t1 < t < t2. This means that, over the small interval ∆t, the angular frequency of c(t) is ωc, which is the slope of the tangential line of θ(t) over this small interval.

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Chapter 3: Frequency Modulation (FM)

For a conventional sinusoid A cos(ωct + θ0), the generalized angle is a straight line ωct + θ0 and the angular frequency is fixed. For a generalizes sinusoid, the angular frequency is not fixed but varies with time. At every time instant t, the instantaneous frequency is the slope of angle θ(t) at time t: ωi(t) = dθ(t) dt The equivalent relationship between angle θ(t) and the instantaneous frequency ωi(t) is: θ(t) = t

−∞

ωi(α)dα

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Chapter 3: Frequency Modulation (FM)

Phase Modulation (PM) and Frequency Modulation (FM)

In PM, the angle θ(t) is varied linearly with the message signal m(t): θ(t) = ωct + kpm(t), (assuming θ0 = 0) sPM(t) = A cos[ωct + kpm(t)], (where kp is a constant) The instantaneous angular frequency ωi(t) of the PM signal is ωi(t) = dθ(t) dt = ωc + kp dm(t) dt , which varies linearly with the derivative of the message. If the instantaneous angular frequency ωi(t) varies linearly with the message, then we have frequency-modulated (FM) signal: ωi(t) = ωc + kfm(t), (where kf is a constant) θ(t) = t

−∞

ωi(α)dα = t

−∞

[ωc + kf m(α)]dα = ωct + kf t

−∞

m(α)dα sFM(t) = A cos

• ωct + kf

t

−∞

m(α)dα

• EE456 – Digital Communications

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Chapter 3: Frequency Modulation (FM)

Relationship Between FM and PM

FM( )

s t

PM( )

s t PM and FM are very much related. It is not possible to tell from the time waveform whether a signal is FM or PM. This is because either m(t), dm(t)

dt

, or

• m(α)dα can

be treated as a message signal.

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Chapter 3: Frequency Modulation (FM)

PM and FM Circuits (Analog)

Note: RFC stands for radio-frequency choke

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Chapter 3: Frequency Modulation (FM)

Example 3

The figure below shows a message signal m(t) and its derivative. Suppose that the constants kf and kp are 2π × 105 and 10π, respectively, and the carrier frequency fc is 100 MHz. (a) Write an expression of the instantaneous frequency of the FM signal. Determine the minimum and maximum values of the instantaneous frequency. (b) Write an expression of the instantaneous frequency of the PM signal. Determine the minimum and maximum values of the instantaneous frequency. (c) Sketch the FM and PM signals and offer your comments.

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Chapter 3: Frequency Modulation (FM)

Solution: (a) For FM, we have: fi(t) = ωi(t) 2π = fc + kf 2π m(t) = 108 + 105m(t) [fi(t)]min = 108 + 105[m(t)]min = 99.9 MHz [fi(t)]max = 108 + 105[m(t)]max = 100.1 MHz (b) For PM, we have: fi(t) = ωi(t) 2π = fc + kp 2π ˙ m(t) = 108 + 5 ˙ m(t) [fi(t)]min = 108 + 5[ ˙ m(t)]min = 99.9 MHz [fi(t)]max = 108 + 5[ ˙ m(t)]max = 100.1 MHz

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Chapter 3: Frequency Modulation (FM)

(c) Sketches of the FM and PM signals are shown below.

FM( )

s t

PM( )

s t Observations: Because m(t) increases and decreases linearly with time, the instantaneous frequency of the FM signal increases linearly from 99.9 to 100.1 MHz over a half-cycle, and then decreases linearly from 100.1 MHz to 99.9 MHz over the remaining half-cycle. Because ˙ m(t) switches back and forth from a value of −20, 000 to 20, 000, the carrier frequency switches back and forth from 99.9 to 100.1 MHz every half-cycle of ˙ m(t).

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Chapter 3: Frequency Modulation (FM)

Comparison of AM, FM and PM Signals with the same massage m(t)

Can you tell which signals on the right are AM, FM and PM, respectively? 5 10 −0.5 0.5 t Message m(t) 5 10 −1 1 t

dm(t) dt

5 10 −0.5 0.5 1 t t

−∞ m(α)dα

5 10 −2 2 t 5 10 −2 2 t 5 10 −2 2 t

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Chapter 3: Frequency Modulation (FM)

Comparison of AM, FM and PM Signals with the same massage m(t)

5 10 −0.5 0.5 t Message m(t) 5 10 −1 1 t

dm(t) dt

5 10 −0.5 0.5 1 t t

−∞ m(α)dα

5 10 −2 2 t sAM(t) 5 10 −2 2 t sPM(t) 5 10 −2 2 t sFM(t)

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Chapter 3: Frequency Modulation (FM)

Comparison of AM, FM and PM Signals under the same amount of noise

Compared to AM, FM and PM signals are much less susceptible to additive noise and

• interference. This is because of two reasons: (i) Additive noise/interference acts on

amplitude, and (ii) the message is embedded in amplitude in AM, while is is embedded in frequency/phase in FM/PM.

1 2 3 4 5 −0.5 0.5 t Message m(t) 1 2 3 4 5 −1 1 t

dm(t) dt

1 2 3 4 5 −0.5 0.5 1 t t

−∞ m(α)dα

1 2 3 4 5 −2 2 t sAM(t) 1 2 3 4 5 −2 2 t sPM(t) 1 2 3 4 5 −2 2 t sFM(t)

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Chapter 3: Frequency Modulation (FM)

Power and Bandwidth of Angle-Modulated Signals

Since the amplitude of either PM or FM signal is a constant A, the power of an angle-modulated (i.e., PM or FM) signal is always A2/2, regardless of the value

• f kp, kf, and power of m(t).

Unlike AM, angle modulation is nonlinear and hence its spectrum/bandwidth analysis is not as simple as for AM signals. To determine the bandwidth of an FM signal, define a(t) = t

−∞

m(α)dα ˆ sFM(t) = Aej[ωct+kf a(t)] = Aejkf a(t)ejωct ⇒ sFM(t) = ℜ{ˆ sFM(t)} Expanding the exponential ejkf a(t) in power series gives: ˆ sFM(t) = A

• 1 + jkfa(t) −

k2

f

2! a2(t) + · · · + jn kn

f

n! an(t) + · · ·

• ejωct

sFM(t) = ℜ{ˆ sFM(t)} = A

• cos(ωct) − kfa(t) sin(ωct) −

k2

f

2! a2(t) cos(ωct) + k3

f

3! a3(t) sin(ωct) + · · ·

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Chapter 3: Frequency Modulation (FM)

Observations: The FM signal consists of an unmodulated carrier and various amplitude-modulated terms, such as a(t) sin(ωct), a2(t) cos(ωct), a3(t) sin(ωct), etc. Since a(t) is an integral of m(t), if M(f) is band-limited to [−B, B], then A(f) is also band-limited to [−B, B]. The spectrum of a2(t) is the spectrum of A(f) ∗ A(f) (where ∗ is the integral convolution operation) and is band-limited to [−2B, 2B]. Similarly, the spectrum

• f an(t) is band-limited to [−nB, nB].

The spectrum of sFM(t) consists of an unmodulated carrier, plus spectra of a(t), a2(t), . . . , an(t), . . . , centered at ωc. Clearly, the bandwidth of sFM(t) is theoretically infinite! For practical message signals, because n! increases much faster than |kfa(t)|n, we have

kn

f an(t)

n!

≈ 0 for large n. Hence most of the modulated-signal power resides in a finite bandwidth. Carson’s rule for Bandwidth Approximation of an FM Signal (captures 98% of total power): BFM = 2(∆f + B) = 2B(β + 1) where ∆f = kf mmax − mmin 2 · 2π is defined as the peak frequency deviation β = ∆f B is the deviation ratio

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Chapter 3: Frequency Modulation (FM)

Spectral Analysis of Tone FM

When the message m(t) is a sinusoid, namely m(t) = Am cos(ωmt), and with the initial condition a(−∞) = 0, one has a(t) = Am ωm sin(ωmt) β = ∆f B = Amkf ωm ˆ sFM(t) = Ae(jωct+jkf Am/ωm sin(ωmt)) = Ae(jωct+jβ sin(ωmt)) = Aejωct ejβ sin(ωmt) Since ejβ sin(ωmt) is a periodic signal with period T = 2π/ωm, it can be expanded by the exponential Fourier series: ejβ sin(ωmt) =

∞

• n=−∞

Dnejnωmt where Dn = ωm 2π π/wm

−π/ωm

ejβ sin(ωmt)e−jnωmtdt = 1 2π π

−π

ej(β sin x−nx)dx = Jn(β)

• nth-order Bessel function of the first kind

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Chapter 3: Frequency Modulation (FM)

It then follows that ˆ sFM(t) = A

∞

• n=−∞

Jn(β)ej(ωct+nωmt) sFM(t) = A

∞

• n=−∞

Jn(β) cos((ωc + nωm)t) Observations: The tone-modulated FM signal has a carrier component and an infinite number

• f sidebands of frequencies ωc ± ωm, ωc ± 2ωm,. . . ,ωc ± nωm. This is very

different from DSB-SC spectrum of tone-modulated AM signal! The strength of the nth sideband at ωc + nωm is A

2 Jn(β), which quickly

decreases with n. In fact, there are only a finite number of significant sideband spectral lines. In general, Jn(β) is negligible for n > β + 1, hence the bandwidth of tone-modulated FM signal is approximated as: BFM = 2(β + 1)fm = 2(∆f + B)

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Chapter 3: Frequency Modulation (FM)

Plot of Jn(β)

0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6 6.4 6.8 7.2 7.6 8 8.4 8.8 9.2 9.6 10 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

β Jn(β)

J0(β) J1(β) J2(β) J3(β) J4(β) J5(β) J6(β)

Two important properties: J−n(β) = (−1)nJn(β)

∞

• n=−∞

J2

n(β) = 1

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Chapter 3: Frequency Modulation (FM)

Table of Jn(β)

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Chapter 3: Frequency Modulation (FM)

Illustration of Tone FM Spectrum

FM( )

/ 2 S f A

FM( )

/ 2 S f A

FM( )

/ 2 S f A

FM( )

/ 2 S f A

FM( )

/ 2 S f A

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Chapter 3: Frequency Modulation (FM)

Example 4

The figure below shows a message signal m(t) and its derivative. Suppose that the constant kf = 2π × 105. (a) Since m(t) is periodic with a fundamental frequency f0 =

1 2×10−4 , it can be

represented as m(t) = ∞

k=−∞ akej2πkf0t. Show that a0=0 and

ak =

• 4

π2k2 ,

k odd 0, k even (b) Assume that the essential bandwidth of m(t) to be the frequency of its third harmonic, estimate the bandwidth of the FM signal when the modulating signal is m(t). (c) Repeat Part (b) if the amplitude of m(t) is doubled (i.e., if m(t) is multiplied by 2). (d) Repeat Part (b) if m(t) is time-expanded by a factor of 2 (i.e., if the period of m(t) is 4 × 10−4).

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Chapter 3: Frequency Modulation (FM)

Narrow-Band FM (NBFM)

sFM(t) = A

• cos(ωct) − kfa(t) sin(ωct) −

k2

f

2! a2(t) cos(ωct) + k3

f

3! a3(t) sin(ωct) + · · ·

• When kf is very small such that |kfa(t)| ≪ 1, then all higher order terms in the

above expression are negligible, except for the first two terms. We then have a good approximation of an FM signal: sFM(t) ≈ A

• cos(ωct) − kf a(t) sin(ωct)
• (1)

The above approximation is a linear modulation similar to that of an AM signal with the message signal being a(t). Because the bandwidth of a(t) is the same as the bandwidth of m(t), which is B Hz, the bandwidth of the narrowband FM signal in (1) is 2B Hz. It is pointed out that the sideband spectrum for a NBFM signal has a phase shift

• f π/2 with respect to the carrier, whereas the sideband spectrum of an AM

signal is in phase with the carrier. The expression of the NBFM signal in (1) suggests a method of generating a NBFM signal by using a DSB-SC modulator (see Fig. 1-(a) on the next slide). The output of the NBFM modulator in Fig. 1-(a) has some amplitude variations (distortion). Such distortion can be removed by using a hard-limiter and a bandpass filter as shown in Fig. 1-(b). The analysis of Fig. 1-(b) shall be explored in Assignment 2.

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Chapter 3: Frequency Modulation (FM)

( ) m t

∫

∑

2 π ( ) a t cos( )

c

A t ω sin( )

c

A t ω − ( )sin( )

f c

Ak a t t ω − NBFM signal ( )cos[ ( )]

c

A t t t ω ϕ + ( )cos[ ( )]

c

A t t t ω ϕ + 4 cos[ ( )]

ct

t ω ϕ π +

Figure 1: Generating a NBFM signal.

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Chapter 3: Frequency Modulation (FM)

Demodulation of FM Signals

Signal at point b : sFM(t) = A cos

• ωct + kf

t

−∞ m(α)dα

• Signal at point c :

˙ sFM(t) = d dt

• A cos
• ωct + kf

t

−∞

m(α)dα

• =

A[ωc + kfm(t)] sin

• ωct + kf

t

−∞

m(α)dα − π

• Signal at point d : A[ωc + kf m(t)]

Signal at point e : kfm(t)

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Chapter 3: Frequency Modulation (FM)

A Practical (Continuous-Time) Differentiator

Recall that the frequency response of an ideal differentiator is H(f) = j2πf. A differentiator can be approximated by a linear system whose frequency response contains a linear segment of a positive slope. One simple device would be an RC high-pass filter. The RC frequency response is simply H(f) = j2πfRC 1 + j2πfRC ≈ j2πfRC, if 2πfRC ≪ 1. Thus, if the parameter RC is very small such that its product with the carrier frequency ωcRC ≪ 1, the RC filter approximates a differentiator.

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Chapter 3: Frequency Modulation (FM)

FCC FM Standards

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Chapter 3: Frequency Modulation (FM)

FM Stations in Saskatoon

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Chapter 3: Frequency Modulation (FM)

Stereo FM

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Chapter 3: Frequency Modulation (FM)

Review of Discrete-Time Processing of Continuous-Time Signals

( )

c

H jω

ˆ

( )

j d

H e ω

The frequency response of the discrete-time LTI system, Hd(ej ˆ

w) is periodic with

period 2π. Over −π ≤ ˆ w ≤ π, it is simply a frequency-scaled version of Hc(ω): Hd(ej ˆ

w) = Hc (ˆ

ωfs) , −π ≤ ˆ w ≤ π where fs = 1

T is the sampling frequency.

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Chapter 3: Frequency Modulation (FM)

Example: Discrete-Time Low-Pass Filter

( )

c

H jω ( )

c

H jω

ˆ

( )

j d

H e ω

ˆ

( )

j d

H e ω

ˆ ω

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Chapter 3: Frequency Modulation (FM)

Discrete-Time Integrator

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Chapter 3: Frequency Modulation (FM)

An illustration of the backward difference, forward difference, and trapezoid rule for approximating the integral of a continuous-time signal using discrete-time processing.

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Chapter 3: Frequency Modulation (FM)

Realization of discrete-time integrators: (a) A realization of the discrete-time integrator based on the trapezoid rule. (b) A realization of the discrete-time integrator based on the backward

• difference. (c) A rearrangement of (b) to produce the more traditional system block diagram of an

accumulator.

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Chapter 3: Frequency Modulation (FM)

• Freq. Responses: Ideal Integrator, Accumulator, Trapezoid-Rule Integrator

Ideal Integrator: Hideal(ej ˆ

w) = 1 j ˆ w .

Accumulator: Hacc(z) =

1 1−z−1 , Hacc(ej ˆ w) = 1 1−e−j ˆ

w .

Trapezoid-Rule Integrator: Htrap(z) = 0.5 1+z−1

1−z−1 , Htrap(ej ˆ w) = 0.5 1+e−j ˆ

w

1−e−j ˆ

w .

−3 −2 −1 1 2 3 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ˆ ω (radians/sample) Magnitude response −π π −π/2 π/2 |Hideal(ejˆ

ω)|

|Hacc(ejˆ

ω)|

|Htrap(ejˆ

ω)|

The accumulator works very well as a DT integrator, especially for small-bandwidth signals.

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Chapter 3: Frequency Modulation (FM)

Discrete-Time Differentiator

ˆ

( )

j d

H e ω

ˆ

( )

j d

H e ω

ˆ ω

EE456 – Digital Communications 35

Chapter 3: Frequency Modulation (FM)

H(ω) = jω, |ω| ≤ Wc 0,

• therwise

⇒ Hd(ej ˆ

w) =

• j ˆ

ω T ,

|ˆ ω| ≤ WcT 0, WcT < |ˆ ω| ≤ π hd[n] = 1 2π WcT

−WcT

j ˆ ω T ej ˆ

wndˆ

ω = WcT πT cos(WcTn) n − 1 πT sin(WcTn) n2 The impulse response has infinite support ⇒ The discrete-time system is an IIR filter. For the special case of full-bandwidth, i.e., when WcT = π, the impulse response is hd[n] =

• 1

T (−1)n n

, n = 0 0, n = 0 The first few samples of the impulse response for the full-bandwidth differentiator are shown below.

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Chapter 3: Frequency Modulation (FM)

An Approximate Discrete-Time Differentiator

By truncating the impulse response to n = −1, 0, 1, the differentiator consists of the three center coefficients. The output of such a differentiator is y[n] = 1 T (x[n + 1] − x[n − 1]) The above system is non-causal. It can be made causal by introducing a delay of 1 sample: y[n] = 1 T (x[n] − x[n − 2])

[ ] x n

1

z−

1

z− + − ∑ [ ] [ ] [ 2] y n x n x n = − −

approximate a differentiator with a delay of 1 sample

Ignoring the scaling factor

1 T , the impulse response of the above approximate

differentiator is h[n] = δ[n] − δ[n − 2] .

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Chapter 3: Frequency Modulation (FM)

The system function H(z) = 1 − z2 has 2 zeros at 0 and π. The frequency response is H(ej ˆ

w)

= 1 − z−2

• z=ej ˆ

w = 1 − e−j2 ˆ

w

= e−j ˆ

w(ej ˆ w − e−j ˆ w) =

e−j ˆ

w delay of 1 sample

(2j sin ˆ ω

≈ˆ ω for ˆ ω small

) ≈ 2jˆ ωe−j ˆ

w −3 −2 −1 1 2 3 0.5 1 1.5 2 2.5 3 3.5 ˆ ω (radians/sample) Magnitude response −π π −π/2 π/2 |Happrox(ejˆ

ω)|

|Hideal(ejˆ

ω)|

The above length-3 FIR approximation to a differentiator works reasonably well for small-bandwidth signal, about |ˆ ω| ≤ 0.2π

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Chapter 3: Frequency Modulation (FM)

Better Approximations of a Discrete-Time Differentiator

Use a Blackman window (Matlab command blackman) to approximate an ideal differentiator as an FIR filter. 0.1 0.2 0.3 0.4 0.5 0.5 1 1.5 2 2.5 3 frequency (cycles/sample) Frequency response N=3, 7, 11, 15, 19, 23, 27, 31

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Chapter 3: Frequency Modulation (FM)

0.1 0.2 0.3 0.4 0.5 −60 −50 −40 −30 −20 −10 10 frequency (cycles/sample) Frequency response (dB) N=3, 7, 11, 15, 19, 23, 27, 31

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Chapter 3: Frequency Modulation (FM)

Building FM Transmitter and Receiver in Lab # 3

Transmitter

+

FM[ ]

cos( [ ])

c

s n n n ω θ = + +

∑

[ ] [ 1] 2 [ ]

f

n n k m n θ θ π = − + ⋅ ⋅ (cycles/sample)

c

f [ ] m n

f

k

EE456 – Digital Communications 41

Chapter 3: Frequency Modulation (FM)

Receiver

cos[( ) ]

c

n ω ω + ∆ sin[( ) ]

c

n ω ω + ∆

c

f f + ∆

[ ]

c

x n [ ]

s

x n [ ]

c

y n [ ]

s

y n

∑

1

z−

1

z−

1

z−

1

z−

' [

1]

c

y n −

' [

1]

s

y n −

'[

1] n θ − + + + − − − [ 1]

s

y n − [ 1]

c

y n −

FM[ ]

s n

EE456 – Digital Communications 42

Chapter 3: Frequency Modulation (FM)

Analysis of the FM Demodulator

sFM(t) = cos

• ωct + kf

t

−∞

m(α)dα

• = cos [ωct + θ(t)]

sFM[n] = cos [ωcnTs + θ(nTs)] = cos (ˆ ωcn + θ[n]) yc[n] = cos(∆ˆ ωn − θ[n]); where ∆ˆ ω = 2π∆ ˆ f ys[n] = sin(∆ˆ ωn − θ[n]); y′

c[n − 1]

≈ d dtyc(t)

• t=(n−1)Ts

, where yc(t) = cos(∆ωt − θ(t)), ∆ω = ∆ˆ ω Ts = −(∆ω − θ′(t)) sin(∆ωt − θ(t))

• t=(n−1)Ts

= −(∆ˆ ω − θ′[n − 1]) sin(∆ˆ ω(n − 1) − θ[n − 1]) Similarly, y′

s[n − 1]

≈ (∆ˆ ω − θ′[n − 1]) cos(∆ˆ ω(n − 1) − θ[n − 1]) Finally, y′

c[n − 1]ys[n − 1] − y′ s[n − 1]yc[n − 1]

= (θ′[n − 1] − ∆ˆ ω)[cos2(∆ˆ ω(n − 1) − θ[n − 1]) + sin2(∆ˆ ω(n − 1) − θ[n − 1])] = (θ′[n − 1] − ∆ˆ ω) = θ′[n − 1] − ∆ˆ ω In the above ∆ˆ ω is the DC offset due to error in the receiver’s local oscillator, while θ′[n − 1] is proportional to m[n − 1].

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