Angle Modulation Phase and Frequency Modulation
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 phase modulation (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 detector.
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"# 2 t "#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 amplitudes go away if we first hard limit the signals, turning them into fixed-amplitude 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