Angle Phase and

Consider a signal of the form xc (t) = Ac cos(2π fct +φ(t)) where Ac and fc are constants. The envelope is a constant so the message cannot be in the envelope. It must instead lie in the variation of the cosine argument with time. Let θi (t) ! 2π fct +φ(t) be the instantaneous phase. Then

jθi(t) xc (t) = Ac cos(θi (t)) = Ac Re(e ).

θi (t) contains the message and this type of modulation is called angle or exponential modulation. If φ(t) = kp m(t) so that xc (t) = Ac cos(2π fct + kp m(t)) the modulation

is called (PM) where kp is the deviation constant or phase modulation index. Phase and Frequency Modulation Think about what it means to modulate the phase of a cosine. The total argument of the cosine is 2π fct +φ(t), an angle with units of radians (or degrees). When

φ(t) = 0, we simply have a cosine and the angle 2π fct is a linear function of time. Think of this angle as the angle of a phasor rotating at a constant angular velocity. Now add the effect of the phase modulation φ(t). The modulation adds a "wiggle" to the rotating phasor with respect to its position when it is unmodulated. The message is in the variation of the phasor's angle with respect to the constant angular velocity of the unmodulated cosine.

Unmodulated Cosine Modulated Cosine

φ()t 2πf ct 2πfct Phase and Frequency Modulation

The total argument of an unmodulated cosine is θc (t) = 2π fct in which fc is a cyclic frequency. The time derivative of 2π fct is 2π fc . We could also express the argument in radian frequency form as θc (t) = ω ct. Its time derivative is ω c . Therefore one way 1 d of defining the cyclic frequency of an unmodulated cosine is as (θc (t)). Now let's 2π dt apply this same idea to a modulated cosine whose argument is θc (t) = 2π fct +φ(t). d Its time derivative is 2π fc + φ(t) . Now we define instantaneous frequency as dt ( ) 1 d 1 ⎡ d ⎤ 1 d f(t) ! (θc (t)) = 2π fc + (φ(t)) = fc + (φ(t)). It is important to draw 2π dt 2π ⎣⎢ dt ⎦⎥ 2π dt a distinction between instantaneous frequency f(t) and spectral frequency f . They are definitely not the same. Let xc (t) = cos(2π fct +φ(t)). It has a Fourier transform Xc ( f ). 1 d Spectral frequency f is the independent variable in Xc ( f ) but f(t) = fc + (φ(t)). 2π dt Some Fourier transforms of phase and frequency modulated signals later will make this distinction clearer. Phase and Frequency Modulation

If we make the variation of the instantaneous frequency of a sinusoid be directly proportional to the message we are doing frequency modulation (FM). If dφ = k f m(t) then k f is the frequency deviation constant in radians/second per dt

k f unit of m(t). In frequency modulation f(t) = fc + fd m(t) , where fd = is 2π the frequency deviation constant in Hz per unit of m(t). In frequency modulation t t φ t = k m λ dλ + φ t = 2π f m λ dλ + φ t , t ≥ t ( ) f ∫ ( ) ( 0 ) d ∫ ( ) ( 0 ) 0 t0 t0 therefore ⎛ t ⎞ x t = A cos 2π f t + 2π f m λ dλ + φ t . c ( ) c ⎜ c d ∫ ( ) ( 0 )⎟ ⎝ t0 ⎠ So PM and FM are very similar. The difference is between integrating the message signal before phase modulating or not integrating it. Phase and Frequency Modulation

Phase−Modulating Signal ) t (

m 0 x

0 2 4 6 8 10 12 14 16 18 20 Time, t (μs)

Phase-Modulated Signal - kp = 5 1 ) t ( c 0 x

−1 0 2 4 6 8 10 12 14 16 18 20 Time, t (μs) Phase in Radians 200 ) t ( c 100 θ

0 0 2 4 6 8 10 12 14 16 18 20 Time, t (μs) Instantaneous Frequency in MHz ) t

f( 1

0 2 4 6 8 10 12 14 16 18 20 Time, t (μs) Phase and Frequency Modulation

Phase−Modulating Signal 1 ) t (

m 0 x

−1 0 2 4 6 8 10 12 14 16 18 20 Time, t (μs)

Phase-Modulated Signal - kp = 5 1 ) t ( c 0 x

−1 0 2 4 6 8 10 12 14 16 18 20 Time, t (μs) Phase in Radians 200 ) t ( c 100 θ

0 0 2 4 6 8 10 12 14 16 18 20 Time, t (μs) Instantaneous Frequency in MHz 1.5 ) t

f( 1

0.5 0 2 4 6 8 10 12 14 16 18 20 Time, t (μs) Phase and Frequency Modulation

Frequency−Modulating Signal 1 ) t (

m 0 x

−1 0 2 4 6 8 10 12 14 16 18 20 Time, t (μs)

Frequency-Modulated Signal - fd = 500,000 1 ) t ( c 0 x

−1 0 2 4 6 8 10 12 14 16 18 20 Time, t (μs) Phase in Radians 200 ) t ( c 100 θ

0 0 2 4 6 8 10 12 14 16 18 20 Time, t (μs) Instantaneous Frequency in MHz 1.5 ) t

f( 1

0.5 0 2 4 6 8 10 12 14 16 18 20 Time, t (μs) Phase and Frequency Modulation Frequency−Modulating Signal 2 ) t (

m 0 x

−2 0 2 4 6 8 10 12 14 16 18 20 Time, t (μs)

Frequency-Modulated Signal - fd = 500,000 1 ) t ( c 0 x

−1 0 2 4 6 8 10 12 14 16 18 20 Time, t (μs) Phase in Radians 200 ) t ( c 100 θ

0 0 2 4 6 8 10 12 14 16 18 20 Time, t (μs) Instantaneous Frequency in MHz 2 ) t

f( 1

0 0 2 4 6 8 10 12 14 16 18 20 Time, t (μs) Phase and Frequency Modulation

Frequency-Modulated Signal - fd = 500,000 1 ) t ( c 0 x

−1 0 2 4 6 8 10 12 14 16 18 20 Time, t (μs)

Instantaneous Frequency in MHz 1.5 ) t

f( 1

0.5 0 2 4 6 8 10 12 14 16 18 20 Time, t (μs)

|X ( f )| c 0.2

0.15

0.1

0.05

f (MHz) −3 −2 2 3 Phase and Frequency Modulation

x(t) Messagex(t) Message

t t

Phase-Modulated Carrier Frequency-Modulated Carrier xc ()t xc ()t

t t Phase and Frequency Modulation

x(t) Message x(t) Message

t t

xc ()t Phase-Modulated Carrier xc ()t Frequency-Modulated Carrier

t t Phase and Frequency Modulation

For phase modulation xc (t) = Ac cos(2π fct + kp m(t)) ⎛ t ⎞ For frequency modulation x t = A cos 2π f t + 2π f m λ dλ c ( ) c ⎜ c d ∫ ( ) ⎟ ⎝ t0 ⎠ There is no simple expression for the Fourier transforms of these signals in the general case. Using cos(x+y) = cos(x)cos(y) − sin(x)sin(y) we can write for PM x t A ⎡cos 2 f t cos k m t sin 2 f t sin k m t ⎤ c ( ) = c ⎣ ( π c ) ( p ( )) − ( π c ) ( p ( ))⎦ ⎡ ⎛ t ⎞ ⎛ t ⎞ ⎤ and for FM x t = A cos 2π f t cos 2π f m λ dλ − sin 2π f t sin 2π f x λ dλ c ( ) c ⎢ ( c ) ⎜ d ∫ ( ) ⎟ ( c ) ⎜ d ∫ ( ) ⎟ ⎥ ⎣⎢ ⎝ t0 ⎠ ⎝ t0 ⎠ ⎦⎥

(under the assumption that φ(t0 ) = 0). Phase and Frequency Modulation

If kp and fd are small enough, cos(kp m(t)) ≅ 1 and sin(kp m(t)) ≅ kp m(t) ⎛ t ⎞ ⎛ t ⎞ t and cos 2π f m λ dλ ≅ 1 and sin 2π f m λ dλ ≅ 2π f m λ dλ. ⎜ d ∫ ( ) ⎟ ⎜ d ∫ ( ) ⎟ d ∫ ( ) ⎝ t0 ⎠ ⎝ t0 ⎠ t0

Then for PM xc (t) ≅ Ac ⎣⎡cos(2π fct) − kp m(t)sin(2π fct)⎦⎤ ⎡ t ⎤ and for FM x t ≅ A cos 2π f t − 2π f sin 2π f t m λ dλ c ( ) c ⎢ ( c ) d ( c )∫ ( ) ⎥ ⎣⎢ t0 ⎦⎥ These approximations are called narrowband PM and narrowband FM. Phase and Frequency Modulation

If the information signal is a sinusoid m(t) = Am cos(2π fmt) then M( f ) = (Am / 2)⎣⎡δ ( f − fm ) +δ ( f + fm )⎦⎤ and, in the narrowband approximation, For PM, xc (t) ≅ Ac ⎣⎡cos(2π fct) − kp Am cos(2π fmt)sin(2π fct)⎦⎤ ⎧ ⎫ ⎪ jAmkp ⎡δ ( f + fc − fm ) +δ ( f + fc + fm ) ⎤⎪ Xc ( f ) ≅ (Ac / 2)⎨⎡δ ( f − fc ) +δ ( f + fc )⎤ − ⎢ ⎥⎬ ⎣ ⎦ 2 f f f f f f ⎩⎪ ⎣⎢−δ ( − c − m ) −δ ( − c + m )⎦⎥⎭⎪ For FM,

⎡ 2π fd Am ⎤ xc (t) ≅ Ac ⎢cos(2π fct) − sin(2π fct)sin(2π fmt)⎥ ⎣ 2π fm ⎦ ⎧ ⎡δ f + f − f −δ f + f + f ⎤⎫ ⎪ Am fd ( c m ) ( c m ) ⎪ Xc ( f ) ≅ (Ac / 2)⎨⎡δ ( f − fc ) +δ ( f + fc )⎤ − ⎢ ⎥⎬ ⎣ ⎦ 2 f f f f f f f ⎩⎪ m ⎣⎢−δ ( − c − m ) +δ ( − c + m )⎦⎥⎭⎪ Phase and Frequency Modulation

Unmodulated Carrier, A = 1, f = 100000000 c c 1 )

t 0.5 c ω 0 cos( c

A -0.5

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -6 Time, t (s) x 10 k ) Quadrature Component, A = 1, f = 1000000, p = 0.1

t m m c

ω 0.1

)sin( 0.05 t m

ω 0 cos(

m -0.05 A p k c -0.1 A

- 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -6 Time, t (s) x 10 Narrowband PM 2

1

0

-1

-2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -6 Time, t (s) x 10

8 x 10 Instantaneous Frequency 1.002

1.001 ) t ( 1 PM f 0.999

0.998 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -6 Time, t (s) x 10 Phase and Frequency Modulation

Unmodulated Carrier, A = 1, f = 100000000 c c 1 )

t 0.5 c ω 0 cos( c

A -0.5

-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -6 Time, t (s) x 10 ) t

c Quadrature Component, A = 1, f = 1000000, f = 100000 m m d ω 0.1 )sin( t

m 0.05 ω 0 )sin( m f / -0.05 m d fA

c -0.1

A 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-( -6 Time, t (s) x 10 Narrowband FM 2

1

0

-1

-2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -6 Time, t (s) x 10

8 x 10 Instantaneous Frequency 1.002

1.001 ) t ( 1 FM f 0.999

0.998 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -6 Time, t (s) x 10 Phase and Frequency Modulation Narrowband PM and FM Spectra for Tone Modulation

Tone-Modulated PM Tone-Modulated FM

Xc ()f Xc ()f

f f -fc fc -fc fc -fc -fm -fc+fm fc -fm fc+fm -fc -fm -fc+fm fc -fm fc+fm

! Xc ()f ! Xc ()f π π

f f

-π -π Phase and Frequency Modulation

If the information signal is a sinc, x(t) = sinc(2Wt) then X( f ) = (1/ 2W )rect( f / 2W ) and, in the narrowband approximation, For PM,

⎧ kp ⎫ X f A / 2 f f f f j ⎡rect f f / 2W rect f f / 2W ⎤ c ( ) ≅ ( c )⎨⎣⎡δ ( − c ) +δ ( + c )⎦⎤ − ⎣ (( + c ) ) − (( − c ) )⎦⎬ ⎩ 2W ⎭ For FM, ⎧ ⎡ ⎤⎫ ⎪ fmk f rect(( f + fc ) / 2W ) rect(( f − fc ) / 2W ) ⎪ Xc ( f ) ≅ (Ac / 2)⎨⎡δ ( f − fc ) +δ ( f + fc )⎤ − ⎢ − ⎥⎬ ⎣ ⎦ 2W f + f f − f ⎩⎪ ⎣⎢ c c ⎦⎥⎭⎪

Phase and Frequency Modulation Narrowband PM and FM Spectra for a Sinc Message

Sinc-Modulated PM Sinc-Modulated FM

Xc ()f Xc ()f

f f -fc -fc − fc − W − fc + W fc − W fc fc + W − fc − W − fc + W fc − W fc fc + W

! Xc ()f ! Xc ()f π π

f f

-π -π Phase and Frequency Modulation Phase and Frequency Modulation If the narrowband approximation is not adequate we must deal with the more complicated wideband case. In the case of tone modulation we can handle PM and FM with basically the same analysis technique if we use the following conventions:

⎪⎧Am sin(2π fmt) , PM x(t) = ⎨ ⎩⎪Am cos(2π fmt) , FM t t For FM, φ t = 2π f x λ dλ = 2π f A cos 2π f λ dλ ( ) d ∫ ( ) d ∫ m ( m ) t0 t0

Am Am φ(t) = 2π fd sin(2π fmt) = fd sin(2π fmt) ω m fm

⎪⎧kp Am , PM Then, for PM and FM, φ(t) = β sin(2π fmt) , where β ! ⎨ ⎩⎪(Am / fm ) fd , FM Then x t = A ⎡cos β sin 2π f t cos 2π f t − sin β sin 2π f t sin 2π f t ⎤ c ( ) c ⎣ ( ( m )) ( c ) ( ( m )) ( c )⎦ Phase and Frequency Modulation In x t A ⎡cos sin 2 f t cos 2 f t sin sin 2 f t sin 2 f t ⎤ c ( ) = c ⎣ ( β ( π m )) ( π c ) − (β ( π m )) ( π c )⎦ cos(β sin(2π fmt)) and sin(β sin(2π fmt)) are periodic with fundamental period 1/ fm . We can now use two results from applied mathematics (Abramowitz and Stegun, page 361) ∞ ∞ cos(zsin(θ )) = J0 (z) + 2∑ J2k (z)cos(2kθ ) = J0 (z) + 2 ∑ Jk (z)cos(kθ ) k=1 k=1 k even ∞ ∞ sin(zsin(θ )) = 2∑ J2k+1 (z)sin((2k +1)θ ) = 2 ∑ Jk (z)sin(kθ ) k=0 k=1 k odd Adapting them to our case ∞ cos(β sin(2π fmt)) = J0 (β ) + 2 ∑ Jk (β )cos(2kπ fmt) k=1 k even ∞ sin(zsin(2π fmt)) = 2 ∑ Jk (β )sin(2kπ fmt) k=1 k odd Phase and Frequency Modulation

⎧⎡ ⎤ ⎡ ⎤ ⎫ ⎪ ∞ ∞ ⎪ x t = A ⎢J β + 2 J β cos 2kπ f t ⎥cos 2π f t − ⎢2 J β sin 2kπ f t ⎥sin 2π f t c ( ) c ⎨⎢ 0 ( ) ∑ k ( ) ( m )⎥ ( c ) ⎢ ∑ k ( ) ( m )⎥ ( c )⎬ ⎪ k=1 k=1 ⎪ ⎩⎣⎢ k even ⎦⎥ ⎣⎢ k odd ⎦⎥ ⎭ ⎧ ∞ ⎫ ⎪J0 (β )cos(2π fct) + 2 ∑ Jk (β )cos(2π fct)cos(2kπ fmt)⎪ k=1 ⎪ k even ⎪ x t A c ( ) = c ⎨ ∞ ⎬ ⎪ ⎪ −2 ∑ Jk (β )sin(2π fct)sin(2kπ fmt) ⎪ k=1 ⎪ ⎩⎪ k odd ⎭⎪ ⎧ ∞ ⎫ J cos 2 f t J ⎡cos 2 f kf t cos 2 f kf t ⎤ ⎪ 0 (β ) ( π c ) + ∑ k (β )⎣ ( π ( c − m ) ) + ( π ( c + m ) )⎦⎪ k=1 ⎪ k even ⎪ x t A c ( ) = c ⎨ ∞ ⎬ ⎪ J ⎡cos 2 f kf t cos 2 f kf t ⎤ ⎪ − ∑ k (β )⎣ ( π ( c − m ) ) − ( π ( c + m ) )⎦ ⎪ k=1 ⎪ ⎩⎪ k odd ⎭⎪ ∞ This can also be written in the more compact form, xc (t) = Ac ∑ Jk (β )cos(2π ( fc + kfm )t) k=−∞ Phase and Frequency Modulation Phase and Frequency Modulation

Ac = 1 , fc = 10 MHz = = 5 = × 5 ⇒β= 7 Am 1 , fm 10 , fd 2 10 2 0.2239cos() 2 ×10 πt 1 J0 ()β

7 0.8 7 −0.5767cos() 1.96 ×10 πt ()β 0.5767cos 2.04 ×10 πt J1 () β β 0.6 J2 () J2 () J ()β J ()β 4 3 J4 ()β 7 ()β 7 0.3528cos() 1.92 ×10 πt 0.4 J5 0.3528cos() 2.08 ×10 πt

0.2

7 β 7 −0.1289cos 1.88 ×10 πt () × π ()k 0.1289cos() 2.12 10 t

J 0

7 0.0340cos() 1.84 ×10 πt -0.2 0.0340cos() 2.16 ×107 πt -0.4 β J5 () β 7 J3 ()J1 ()β 7 −0.007cos() 1.8 ×10 πt -0.6 0.007cos() 2.2×10 πt -10 -5 0 5 10 β β=2 Phase and Frequency Modulation

Ac = 1 , fc = 10 MHz 5 5 Am = 1 , fm = 10 , fd = 2 ×10 ⇒β=2 1 J0 (β) 0.8 J1 (β) β β 0.6 J2 ( ) J2 ( ) J (β) J (β) 4 3 J4 (β) β 0.4 J5 ( )

0.2 β () k

J 0

-0.2

-0.4 β J5 ( ) β J3 ( ) J1 (β) -0.6 -10 -5 0 5 10 β β=−2 β=2

f (MHz) 9 9.2 9.4 9.6 9.8 10 10.2 10.4 10.6 10.8 11 Phase and Frequency Modulation

Now, to find the spectrum of xc (t) take the Fourier transform of xc (t).

∞ X f A / 2 J ⎡ f f kf f f kf ⎤ c ( ) = ( c ) ∑ k (β)⎣δ ( − ( c + m )) + δ ( + ( c + m ))⎦ k=−∞

The impulses in the spectrum extend in frequency all the way to infinity. But beyond

β fm the impulse strengths die rapidly. For practical purposes the bandwidth is approximately 2β fm . Phase and Frequency Modulation Wideband FM Spectrum for Cosine-Wave Modulation

Xc ()f β = 8

β fm

f f ! c Xc ()f π

f Phase and Frequency Modulation

Tone PM, β = 5, φ = 5, A = 1, f = 1 Tone FM, β = 5, f = 5, A = 1, f = 1 ΔΔm m m m Tone PM, β = 10, φ = 5, A = 2, f = 1 Tone FM, β = 10, f = 5, A = 2, f = 1 ΔΔm m m m |X ( f )| |X ( f )| |X ( f )| |X ( f )| c,PM c,FM c,PM c,FM 0.2 0.2 0.2 0.2 0.15 0.15 0.15 0.15 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 f f f f 20 40 60 80 20 40 60 80 20 40 60 80 20 40 60 80

Tone PM, β = 5, φ = 5, A = 1, f = 2 Tone FM, β = 2.5, f = 5, A = 1, f = 2 Tone PM, β = 10, φ = 5, A = 2, f = 2 Tone FM, β = 5, f = 5, A = 2, f = 2 ΔΔm m m m Δ m m Δ m m |X ( f )| |X ( f )| |X ( f )| |X ( f )| c,PM c,FM c,PM c,FM 0.2 0.25 0.2 0.2 0.15 0.2 0.15 0.15 0.15 0.1 0.1 0.1 0.1 0.05 0.05 0.05 0.05 -50 -50 -50 -50 f f f f 50 100 150 50 100 150 50 100 150 50 100 150 Transmission Bandwidth

The bandwidth required for transmitting an FM signal is theoretically infinite. That is, an infinite bandwidth would be required to transmit an FM signal perfectly, even if the modulating signal is bandlimited. Fortunately, in practical systems, perfection is not required and we can get by with a finite bandwidth. With tone modulation, the bandwidth required depends on the modulation index β. The spectral line magnitudes fall off rapidly at positive frequencies for which f − fc > β fm . So for tone modulation the bandwidth required for transmission would be approximately 2β fm . In the narrowband case when β is very small we cannot exactly follow this rule because we would have no modulation at all. So there is a "floor" of at least 2fm. Transmission Bandwidth

For the general case, Carson's rule is a handy approximation that says B ≅ 2(D +1)W , where peak frequency deviation f D = = d m(t) bandwidth of m(t) W max

If D << 1, then B ≅ 2W . This is the narrowband case.

If D 1, then B 2DW f m t . This is the wideband case. >> ≅ = d ( ) max Generation and Detection of FM and PM The most direct and straightforward way of generating FM is to use a device known as a voltage-to-frequency converter (VCO). One way this can be done is by varying with time the capacitance in an LC parallel resonant oscillator. Let the capacitance be the capacitance of a varactor diode in parallel with another capacitor forming C(t) = C0 − C x(t). The time-varying LC resonant frequency is 1 d d 1 1 1 f(t) = (θ (t)) ⇒ (θ (t)) = = 2π dt dt L C(t) LC0 C 1− x(t) C0 We can use the formula (Abramowitz and Stegun, page 15),

α α (α −1) 2 α (α −1)(α − 2) 3 (1+ x) = 1+ αx + x + x +! 2! 3! to write

−1/2 2 d 1 ⎡ C ⎤ 1 ⎡ 1 C 3 ⎛ C ⎞ ⎤ (θ (t)) = ⎢1− x(t)⎥ = ⎢1+ x(t) + x(t) +!⎥ dt LC C LC 2 C 8 ⎝⎜ C ⎠⎟ 0 ⎣ 0 ⎦ 0 ⎣⎢ 0 0 ⎦⎥ Generation and Detection of FM and PM d 1 ⎡ 1 C ⎤ If C x(t) is "small enough", then (θ (t)) ≅ ⎢1+ x(t)⎥ and dt LC0 ⎣ 2 C0 ⎦ C t C θ t = 2π f t + 2π f x λ dλ. This is in the form of FM with f = f . ( ) c c ∫ ( ) d c 2C0 2C0

Since x(t) ≤ 1, the approximation is good to within one percent if C / C0 < 0.013. C So, taking that as an upper limit, fd = fc ≤ 0.006 fc . This is a practical result 2C0 that usually causes no design problems.

Tuned DC Block N :1 RFC Circuit + +

x()t L xC ()t Oscillator Cv ()t C1 − Varactor VB − Generation and Detection of FM and PM

Another method for generating FM is to use a phase modulator, which produces PM, but integrate the message before applying it to the phase modulator. A narrowband phase modulator can be made by simulating the narrowband approximation xc (t) = Ac cos(2π fct) − Ackp x(t)sin(2π fct).

k p x()t xc ()t

+90°

−Ac sin()2πfc t

Ac cos()2πfc t Generation and Detection of FM and PM

A third method for generating FM is called indirect FM. First, integrate 1 t the message x(t). Then use the integral of the message x(λ)dλ as the T ∫ input signal to a narrowband phase modulator with a carrier frequency fc1. This

kp produces a signal with instantaneous frequency f1 (t) = fc1 + x(t). 2πT

Narrowband Frequency Modulator

f ()t Frequency f ()t f()t RF 1 Phase 1 2 x()t Multiplier Power xc ()t T ∫ Modulator ×n Amp

fLO fc1 Generation and Detection of FM and PM Next frequency-multiply the narrowband FM signal by a factor of n. This moves the carrier frequency to nfc1, creating a signal with instantaneous frequency

kp f2 (t) = nfc1 + n x(t). The effective value of the frequency deviation is now 2πT k f = n p . This changes the range of frequency variation but not the rate of d 2πT frequency variation. Then, if needed, shift the entire FM spectrum to whatever carrier frequency is required and amplify for transmission.

Narrowband Frequency Modulator

f ()t Frequency f ()t f()t RF 1 Phase 1 2 x()t Multiplier Power xc ()t T ∫ Modulator ×n Amp

fLO fc1 Generation and Detection of FM and PM There are four common methods of detecting FM: 1. FM-to-AM Conversion Followed by Envelope Detection 2. Phase-Shift Discrimination 3. Zero-Crossing Detection 4. Frequency Feedback FM-to-AM conversion can be done by time-differentiating the modulated signal. ! Let xc (t) = Ac cos(θc (t)) with θ (t) = 2π ⎣⎡ fc + fd x(t)⎦⎤. Then ! x! c (t) = −Acθ (t)sin(θc (t)) = 2π Ac ⎣⎡ fc + fd x(t)⎦⎤sin(θc (t) ±180°). The message can then be recovered by an envelope .

Envelope DC x ()t Limiter LPF d/dt yD ()t c Detector Block Generation and Detection of FM and PM The "differentiator" in FM-to-AM detection need not be a true differentiator. All that is really needed is a frequency response magnitude that has a linear (or almost linear) slope over the bandwidth of the FM signal. Just below and just above resonance a tuned circuit resonator has an almost linear magnitude dependence on frequency. This type of detection is commonly called slope detection.

|H()f | Almost Linear Slope

f

fc f0 Generation and Detection of FM and PM

The linearity of slope detection can be improved by using two resonant circuits instead of only one. This type of circuit is called a balanced discriminator.

> $f0 %&fc +

+ C L

f xC ()t K x()t fc − C L

− !"#

f0 < fc Phase and Frequency

Consider a cosine of the form x(t) = Acos(2π f0t +φ(t)). The phase of this cosine is θ (t) = 2π f0t +φ(t) and φ(t) is its phase shift.

x t # Acos t First consider the case φ t = 0. !" !"0 ( ) A Then x t Acos 2 f t ( ) = ( π 0 ) t and θ (t) = 2π f0t. T0

T0 # 1/ f0 # 2 / 0

The cyclic frequency of this cosine is f0 . Also, the first time derivative of θ (t) is 2π f0 . So one way of defining cyclic frequency is as the first derivative of phase, divided by 2π. It then follows that phase is the integral of frequency. Phase and Frequency

If x(t) = Acos(2π f0t) and θ (t) = 2π f0t. Then a graph of phase versus time would be a straight line through the origin with slope 2π f0 .

x!"t # Acos!" 0t A

t

T0

T0 # 1/ f0 # 2 / 0

θ ()t

2πf0 t 1 Phase and Frequency

Let x(t) = Acos(θ (t)) and let θ (t) = 2π f0t u(t) = 2π f0 ramp(t). Then the 1 d cyclic frequency is θ (t) = f0 u(t). Call its instantaneous cyclic 2π dt frequency f(t). x"#t ! Acos"# 0t u"#t A t

T0

T0 ! 1/ f0 ! 2 / 0 f(t) θ ()t 2πf f0 0 t t 1 Phase and Frequency Now let x(t) = Acos(2πt (u(t) + u(t −1))). Then the instantaneous cyclic frequency is f(t) = u(t) + u(t −1) and the phase is θ (t) = 2π (ramp(t) + ramp(t −1)).

x"#t ! Acos"# 2t "#u"#t $ u"#t 1 A t

t ! 1 t ! 2 f ()t θ 2 ()t 1 2π t t 1 1 Phase Discrimination

Let x1 (t) = A1sin(2π f0t +θ (t)) and let x2 (t) = A2 cos(2π f0t +φ(t)).

The product is x1 (t)x2 (t) = A1A2sin(2π f0t +θ (t))cos(2π f0t +φ(t)). Using a trigonometric identity,

A1A2 x1 (t)x2 (t) = ⎡sin φ(t) −θ (t) + sin 4π f0t +φ(t) +θ (t) ⎤ 2 ⎣ ( ) ( )⎦ and

A1A2 x1 (t)x2 (t) = sin φ(t) −θ (t) 2 ( )

A A 1 2 ⎡ φ −θ + π +φ +θ ⎤ ⎣sin()()t ()t sin() 4 f0t ()t ()t ⎦ 2 A1 A2 /2 A A 90° 1 2 φ −θ t t A1sin() 2π f0t +θ()t LPF sin()()t ()t () () 2 90°

A2 cos() 2π f0t +φ()t Voltage-Controlled Oscillators

A voltage - controlled oscillator (VCO) is a device that accepts an analog voltage as its input and produces a periodic waveform whose fundamental frequency depends on that voltage. Another common name for a VCO is "voltage-to-frequency converter". The waveform is typically either a sinusoid or a rectangular wave.

A VCO has a free-running frequency fv. When the input analog voltage is zero, the fundamental frequency of the VCO output signal is fv. The output frequency of the VCO is fVCO = fv + Kv vin where

Kv is a gain constant with units of Hz/V. Phase-Locked Loops A phase - locked loop (PLL) is a device used to generate a signal with a fixed phase relationship to the carrier in a bandpass signal. An essential ingredient in the locking process is an analog phase comparator. A phase comparator produces a signal that depends on the phase difference between two bandpass signals. One system that accomplishes this goal is an analog mulitplier followed by a lowpass filter. Let the two bandpass signals be

xr (t) = Ac cos(2π fct +φ(t)) and e0 (t) = Av sin(2π fct +θ (t)) and let the

output signal from the phase comparator be ed (t).

xr ()t = Ac cos() 2π fct +φ()t LPF -Kd ed(t )

e0 ()t = Av sin() 2π fct +θ ()t Phase-Locked Loops

xr (t)e0 (t) = −Kd Ac Av cos(2π fct +φ(t))sin(2π fct +θ (t))

Kd Ac Av xr (t)e0 (t) = − ⎡sin θ (t) −φ(t) + sin 4π fct +φ(t) +θ (t) ⎤ 2 ⎣ ( ) ( )⎦

Kd Ac Av Kd Ac Av ed (t) = sin φ(t) −θ (t) = sin ψ (t) 2 ( ) 2 ( )

ed ()t Kd Ac Av 2

90° 90° xr ()t = Ac cos() 2π fct +φ()t LPF -Kd ed(t )

e0 ()t = Av sin() 2π fct +θ ()t Phase-Locked Loops

ed (t) depends on both the phase difference and Ac and Av. We can make the dependence on these go away if we first hard limit the signals, turning them into fixed- square waves. Another benefit of hard- limiting is that the multiplication becomes a switching operation and the error

signal ed (t) is now a linear function of ψ (t) over a wider range.

Hard Switching xr ()t LPF ed()t Limiter Circuit ed ()t

90° Hard e0()t 90 Limiter ° Phase-Locked Loops From the block diagram of the phase-locked loop below it is clear that Ev (s) = F(s)Ed (s), where F(s) is the transfer function of the loop-filter-loop-amplifier combination. The VCO converts voltage to frequency and phase is the integral of frequency. That is why the VCO is represented as an integrator with voltage in and phase out.

Phase Comparator

+ ed(t ) Kd Ac Av Loop ()t sin(•) Filter − 2 t ()t ev(t ) Loop Kv Amplifier

VCO Phase-Locked Loops

Phase-locked loops operate in two modes, acquisition and tracking. When a PLL is turned on it must first acquire a phase lock and thereafter it must track the phase changes in the incoming signal. The acquisition of a phase lock must be described by the non-linear model of the PLL in which the phase discriminator has a sine transfer function. In the tracking mode, the phase error is typically small, the sine function can be approximated by its argument and the model of the PLL becomes linear.

Phase-Locked Loops

Phase Comparator Non-linear + ed(t ) Kd Ac Av Loop PLL Model ()t sin(•) Filter 2 for Acquisition − t ()t ev(t ) Loop Kv Amplifier

VCO Phase Comparator

Linear PLL + K A A ed(t ) Loop ()t d c v Model for Filter − 2 Tracking t ()t ev(t ) Loop Kv Amplifier

VCO Phase-Locked Loops K A A K In the tracking mode Θ(s) = d c v ⎡Φ(s) − Θ(s)⎤F(s) v . 2 ⎣ ⎦ s

Θ(s) Kt F(s) Kd Ac AvKv It follows that H(s) = = where Kt = . Φ(s) s + Kt F(s) 2 Ψ(s) Φ(s) − Θ(s) Ψ(s) = Φ(s) − Θ(s), therefore G(s) = = = 1− H(s). Φ(s) Φ(s)

Phase Comparator

+ ed(t ) Kd Ac Av Loop ()t Filter − 2 t ()t ev(t ) Loop Kv Amplifier

VCO Phase-Locked Loops

Let the phase deviation of the incoming signal φ(t) be of the general

2 1 dφ form φ(t) = ⎡π Rt + 2π fΔ (t) +θ0 ⎤u(t). Then = (Rt + fΔ )u(t), ⎣ ⎦ 2π dt 2π R 2π f θ a frequency ramp plus a frequency step. Then Φ(s) = + Δ + 0 . s3 s2 s Using the final value theorem of the Laplace transform, the steady state phase error between the incoming signal and the VCO output signal is 2 ⎡2π R 2π fΔ θ0 ⎤ s + as + b limψ (t) = lim s 3 + 2 + G(s). Now let F(s) = 2 . t→∞ s→0 ⎣⎢ s s s ⎦⎥ s 2 3 Kt (s + as + b) s Then H(s) = 3 2 and G(s) = 3 2 . s + Kt (s + as + b) s + Kt (s + as + b) Phase-Locked Loops θ s2 + 2π f s + 2π R Then the steady-state phase error is limψ (t) = lim s 0 Δ . t→∞ s→0 3 2 s + Kt (s + as + b) Its value depends on the form of the input signal's phase deviation and the order of the loop filter. Steady State Error, limψ (t) t→∞

θ ≠ 0 θ ≠ 0 θ ≠ 0 PLL Order 0 0 0 f = 0 f ≠ 0 f ≠ 0 ↓ Δ Δ Δ R = 0 R = 0 R ≠ 0 2π f 1(a = 0,b = 0) 0 Δ ∞ Kt 2π R 2(a ≠ 0,b = 0) 0 0 t 3(a ≠ 0,b ≠ 0) 0 0 0 Phase-Locked Loops

So a first-order PLL can track a phase step with zero error and a frequency step with a finite error. A second-order PLL can track a frequency step with zero error and a frequency ramp with a finite error. A third-order PLL can track a frequency step and a frequency ramp with zero error. When the error is finite, its size can be made arbitrarily small by making Kt large. However, this also increases the loop bandwidth, making the signal-to-noise ratio worse. (More in Chapter 8.) Phase-Locked Loops A first-order PLL can be used for demodulation of angle-modulated signals but a second-order PLL has some advantages and is more s2 + as + b common in practice. Therefore, in F(s) = , make b = 0. s2 a Then F(s) = 1+ . This can be implemented as the signal plus a times s

Θ(s) Kt (s + a) the integral of the signal. With this F(s), H(s) = = 2 and Φ(s) s + Kt (s + a) Ψ(s) s2 G(s) = = 2 , a second-order transfer function. Expressing Φ(s) s + Kt (s + a) this transfer function in a standard second-order system form,

2 s 1 Kt G(s) = 2 2 , where ζ = is the damping factor and s + 2ζω 0s +ω 0 2 a

ω 0 = Kt a is the radian natural frequency. Phase-Locked Loop States Input and Feedback Signals at Same Frequency

Locked in Quadrature Not Locked - Input and Feedback Signals in Phase

t t t t

= ε t K y()t = Kasin()ε ()t t LPF Ka y()t Kasin()()t LPF a

VCO VCO K Kv v t t t t

Not Locked - Input and Feedback Signals 180° Out of Phase

t t

= ε t LPF Ka y()t Kasin()()t

VCO

Kv t t Phase-Locked Loop States Input and Feedback Signals at Different Frequencies

Input Frequency > Feedback Frequency Input Frequency < Feedback Frequency

t t t t

t LPF Ka t LPF Ka

VCO VCO

Kv Kv t t t t Phase-Locked Loops For DSB signals, which do not have transmitted carriers, Costas invented a system to synchronize a local oscillator and also do

synchronous detection. The incoming signal is xc (t) = x(t)cos(ω ct) with bandwidth 2W . It is applied to two phase discriminators, main and quad, each consisting of a multiplier followed by a LPF and an amplifier. The local oscillators that drive them are 90° out of phase so

that the output signal from the main phase discriminator is x(t)sin(εss )

and the output signal from the quad phase discriminator is x(t)cos(εss ).

x()t sin()ε ss Main PD

t

cos()ω ct −εss + 90° VCO Error Signals T ∫ = ε t −T ω yss Sx sin() 2 ss x()t cos()ct 2 −90°

Quad PD Output

x()t cos()ε ss Phase-Locked Loops

The VCO control voltage yss is the time average of the product of x(t)sin(εss ) t 2 and x(t)cos εss or yss = x (λ)cos εss sin εss dλ which is ( ) ∫t−T ( ) ( )

T 2 T yss = x (t)⎡sin(0) + sin(2εss )⎤ = Sx sin(2εss ). When the angular error 2 ⎣ ⎦ 2

εss is zero, yss does not change with time, the loop is locked and the output

signal from the quad phase discriminator is x(t)cos(εss ) = x(t) because εss = 0.

x()t sin()ε ss Main PD

t

cos()ω ct −εss + 90° VCO Error Signals T ∫ = ε t −T ω yss Sx sin() 2 ss x()t cos()ct 2 −90°

Quad PD Output

x()t cos()ε ss Interference

Let the total received signal at a receiver be

v(t) = Ac cos(ω ct) + Ai cos((ω c + ωi )t + φi ) where the first term represents the desired signal and the second term represents interference. Also define ρ ! Ai / Ac and θi (t) ! ωit + φi . Then ⎧ ⎫ ⎪ ⎡cos(ω ct)cos(θi (t)) ⎤⎪ v(t) = Ac ⎣⎡cos(ω ct) + ρ cos(ω ct +θi )⎦⎤ = Ac ⎨cos(ω ct) + ρ ⎢ ⎥⎬ sin t sin t ⎩⎪ ⎣⎢− (ω c ) (θi ( ))⎦⎥⎭⎪ v t A ⎡1 cos t ⎤cos t sin t sin t ( ) = c {⎣ + ρ (θi ( ))⎦ (ω c ) − ρ (θi ( )) (ω c )}

The in-phase component is Ac ⎣⎡1+ ρ cos(θi (t))⎦⎤cos(ω ct) and the quadrature component is

−Acρsin(θi (t))sin(ω ct). The envelope is

2 2 2 Av (t) = Ac ⎣⎡1+ ρ cos(θi )⎦⎤ + ρ sin(θi (t)) = Ac 1+ ρ + 2ρ cos(θi (t)). The phase relative

−1 ⎛ ρsin(θi (t)) ⎞ to the desired signal is φv (t) = tan . ⎜ 1+ ρ cos θ t ⎟ ⎝ ( i ( ))⎠ Interference

The envelope and phase of the total received signal

2 −1 ⎛ ρsin(θi (t)) ⎞ Av (t) = Ac 1+ ρ + 2ρ cos(θi (t)) and φv (t) = tan ⎜ ⎟ ⎝ 1+ ρ cos(θi (t))⎠ show that the effect of the interference on the received signal is to create both amplitude and phase modulation. If ρ << 1, then

−1 Av (t) ≅ Ac 1+ 2ρ cos(θi (t)) ≅ Ac ⎣⎡1+ ρ cos(θi (t))⎦⎤ and φv (t) ≅ tan (ρsin(θi (t))) ≅ ρsin(θi (t)) or

Av (t) ≅ Ac ⎣⎡1+ ρ cos(ωit + φi )⎦⎤ and φv (t) ≅ ρsin(ωit + φi ) This result has the form of AM tone modulation with µ = ρ and simultaneous PM or FM tone modulation with β = ρ. If ρ >> 1, then

−1 −1 Av (t) = ρAc 1+ 2ρ cos(ωit + φi ) ≅ ρAc ⎣⎡1+ ρ cos(ωit + φi )⎦⎤ and φv (t) = ωit + φi Interference In the weak interference case

Av (t) ≅ Ac ⎣⎡1+ ρ cos(ωit + φi )⎦⎤ and φv (t) ≅ ρsin(ωit + φi ) if we demodulate with an envelope, phase or frequency demodulator we get

(with φi = 0)

Envelope Detector: K D ⎣⎡1+ ρ cos(ωit)⎦⎤

Phase Detector: K Dρsin(ωit)

Frequency Detector: K Dρ fi cos(ωit) For AM or PM demodulation the demodulated signal strength is proportional to ρ. For FM demodulation the demodulated signal strength is porportional to the product of ρ and fi .

FM AM or PM Amplitude f W i Interference

The effects of interference on FM signals increases with frequency. So one way to reduce the effect is to lowpass filter the demodulated output. Of course this also lowpass filters the message, an undesirable outcome. To avoid the lowpass filtering effect on the message a technique called preemphasis is often used. The higher frequency parts of the message are preemphasized before transmission by passing them through a preemphasis filter with frequency response H pe ( f ) that amplifies the higher frequencies more than the lower frequencies. Then, after transmission and frequency demodulation, the demodulated signal is passed through a deemphasis filter 1 whose frequency response is Hde ( f ) = . H pe ( f ) Interference

1 A typical deemphasis filter has a frequency response Hde ( f ) = in 1+ jf / Bde which Bde is less than the cutoff frequency of the normal sharp-cutoff lowpass filter that determines the bandwidth. That makes the corresponding preemphasis filter have a frequency response H pe ( f ) = 1+ jf / Bde. Interference

A phenomenon that most people have experienced in receiving FM signals is the so-called capture effect. Suppose there are two FM stations, both transmitting in the same bandwidth and of approximately equal signal strength at the receiver. Their signal strengths will fluctuate some causing one to be stronger for a time and then the other. The stronger signal will "capture" the receiver for a short time and will dominate the demodulated signal. But then later the other signal will dominate and capture the receiver. The two stations switch back and forth and the listener hears a time-multiplexed version of both signals. To keep the math simple, assume we have one unmodulated carrier and one modulated carrier. This is exactly the "interfering sinusoid" case we analyzed earlier with the results

2 −1 ⎛ ρsin(θi (t)) ⎞ Av (t) = Ac 1+ ρ + 2ρ cos(θi (t)) and φv (t) = tan ⎜ ⎟ ⎝ 1+ ρ cos(θi (t))⎠ with θi (t) = φi (t), the phase modulation of the interfering signal. Interference

2 −1 ⎛ ρsin(θi (t)) ⎞ Av (t) = Ac 1+ ρ + 2ρ cos(θi (t)) and φv (t) = tan ⎜ ⎟ ⎝ 1+ ρ cos(θi (t))⎠ The demodulated signal is then ⎛ ⎞ d −1 ⎛ ρsin(φi (t)) ⎞ yD (t) = φv (t) = ⎜ tan ⎜ ⎟ ⎟ . dt ⎝ ⎝ 1+ ρ cos(φi (t))⎠ ⎠ d 1 Using tan−1 (z) = and the chain rule of differentiation, dz ( ) 1+ z2 ! ! 1 ⎣⎡1+ ρ cos(φi (t))⎦⎤ ρ cos(φi (t))φ(t) + ρsin(φi (t))ρsin(φi (t))φ(t) yD (t) = 2 × 2 ⎛ ⎞ ρsin(φi (t)) ⎣⎡1+ ρ cos(φi (t))⎦⎤ 1+ ⎜ ⎟ ⎝ 1+ ρ cos(φi (t))⎠ ρ cos φ t + ρ2 ( i ( )) ! yD t = φ t ( ) 2 2 2 ( ) ⎣⎡1+ ρ cos(φi (t))⎦⎤ + ρ sin (φi (t)) ρ ⎡ρ + cos φ t ⎤ ⎣ ( i ( ))⎦ ! ! yD (t) = 2 φ(t) = α (ρ,φi )φ(t) 1+ ρ + 2ρ cos(φi (t))

ρ ⎣⎡ρ + cos(φi (t))⎦⎤ where α (ρ,φi ) = 2 1 2 cos t + ρ + ρ (φi ( )) Interference

! yD (t) = α (ρ,φi )φ(t) ! The φ(t) factor suggests that the interference may be intelligible if α (ρ,φi ) ! is relatively constant with time. If ρ >> 1, then α (ρ,φi ) ≅ 1 and yD (t) ≅ φ(t). But we wish to examine the case in which the two signals are approximately equal in strength, implying that ρ ≅ 1.

⎧ρ / (1+ ρ) , φi = 0 + 2nπ ⎡ cos t ⎤ ρ ⎣ρ + (φi ( ))⎦ ⎪ 2 2 α (ρ,φi ) = 2 = ⎨ρ / 1+ ρ , φi = π / 2 + nπ , n an integer 1 2 cos t ( ) + ρ + ρ (φi ( )) ⎪ −ρ / 1− ρ , φ = π + 2nπ ⎩ ( ) i

1 ρ = 0.9 0.5

0 ρ = 0.1

) -0.5 φ ,

ρ -1 (

α -1.5

-2

-2.5

-3 0 1 2 3 4 5 6 φ Interference ρ ⎡ρ + cos φ t ⎤ ! ⎣ ( i ( ))⎦ yD (t) = α (ρ,φi )φ(t) , α (ρ,φi ) = 2 1+ ρ + 2ρ cos(φi (t)) ! As ρ → 1 , α → 0.5 and yD (t) → 0.5φ(t). For ρ < 1, the strength of the demodulated interference depends mostly on the peak-to-peak value of α 2ρ α p− p = α (ρ,0) − α (ρ,π ) = 2 (1− ρ) The interference effect is small-to-negligible for ρ < 0.7 and the interference captures the demodulated output signal when ρ > 0.7. 1 10 ρ = 0.9 0.5 8 0 ρ = 0.1

) -0.5 ) 6 φ ρ , (

ρ -1 p-p ( α α -1.5 4

-2 2 -2.5

-3 0 0 1 2 3 4 5 6 0 0.2 0.4 0.6 0.8 1 φ ρ