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Angle Modulation Phase and Frequency Modulation

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Angle Modulation Phase and Frequency Modulation 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.
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