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. EE456 – Digital Communications 2
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 [ t 1 , t 2 ] , draw a tangential line of θ ( t ) , which can be described by equation ω c t + θ 0 . It is clear from the figure that, over the interval t 1 < t < t 2 one has: c ( t ) = A cos θ ( t ) = A cos( ω c t + θ 0 ) , t 1 < t < t 2 . This means that, over the small interval ∆ t , the angular frequency of c ( t ) is ω c , which is the slope of the tangential line of θ ( t ) over this small interval. EE456 – Digital Communications 3
Chapter 3: Frequency Modulation (FM) For a conventional sinusoid A cos( ω c t + θ 0 ) , the generalized angle is a straight line ω c t + θ 0 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 : ω i ( t ) = d θ ( t ) d t The equivalent relationship between angle θ ( t ) and the instantaneous frequency ω i ( t ) is: � t θ ( t ) = ω i ( α )d α −∞ EE456 – Digital Communications 4
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 ) = ω c t + k p m ( t ) , ( assuming θ 0 = 0) s PM ( t ) = A cos[ ω c t + k p m ( t )] , ( where k p is a constant ) The instantaneous angular frequency ω i ( t ) of the PM signal is ω i ( t ) = d θ ( t ) d m ( t ) = ω c + k p , d t d t which varies linearly with the derivative of the message . If the instantaneous angular frequency ω i ( t ) varies linearly with the message , then we have frequency-modulated (FM) signal: ω i ( t ) = ω c + k f m ( t ) , ( where k f is a constant ) � t � t � t θ ( t ) = ω i ( α )d α = [ ω c + k f m ( α )]d α = ω c t + k f m ( α )d α −∞ −∞ −∞ � t � � s FM ( t ) = A cos ω c t + k f m ( α )d α −∞ EE456 – Digital Communications 5
Chapter 3: Frequency Modulation (FM) Relationship Between FM and PM s FM ( ) t s PM ( ) 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 ) , d m ( t ) � , or m ( α )d α can d t be treated as a message signal. EE456 – Digital Communications 6
Chapter 3: Frequency Modulation (FM) PM and FM Circuits (Analog) Note: RFC stands for radio-frequency choke EE456 – Digital Communications 7
Chapter 3: Frequency Modulation (FM) Example 3 The figure below shows a message signal m ( t ) and its derivative. Suppose that the constants k f and k p are 2 π × 10 5 and 10 π , respectively, and the carrier frequency f c 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. EE456 – Digital Communications 8
Chapter 3: Frequency Modulation (FM) Solution: (a) For FM, we have: ω i ( t ) = f c + k f 2 π m ( t ) = 10 8 + 10 5 m ( t ) f i ( t ) = 2 π 10 8 + 10 5 [ m ( t )] min = 99 . 9 MHz [ f i ( t )] min = 10 8 + 10 5 [ m ( t )] max = 100 . 1 MHz [ f i ( t )] max = (b) For PM, we have: ω i ( t ) = f c + k p m ( t ) = 10 8 + 5 ˙ f i ( t ) = 2 π ˙ m ( t ) 2 π 10 8 + 5[ ˙ [ f i ( t )] min = m ( t )] min = 99 . 9 MHz 10 8 + 5[ ˙ [ f i ( t )] max = m ( t )] max = 100 . 1 MHz EE456 – Digital Communications 9
Chapter 3: Frequency Modulation (FM) (c) Sketches of the FM and PM signals are shown below. s FM ( ) t s PM ( ) 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 ) . EE456 – Digital Communications 10
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? 0.5 2 Message m ( t ) 0 0 −0.5 −2 0 5 10 0 5 10 t t 1 2 d m ( t ) d t 0 0 −1 −2 0 5 10 0 5 10 t t 1 2 − ∞ m ( α )d α 0.5 0 0 � t −0.5 −2 0 5 10 0 5 10 t t EE456 – Digital Communications 11
Chapter 3: Frequency Modulation (FM) Comparison of AM, FM and PM Signals with the same massage m ( t ) 0.5 2 Message m ( t ) s AM ( t ) 0 0 −0.5 −2 0 5 10 0 5 10 t t 1 2 s PM ( t ) d m ( t ) 0 0 d t −1 −2 0 5 10 0 5 10 t t 1 2 − ∞ m ( α )d α 0.5 s FM ( t ) 0 0 � t −0.5 −2 0 5 10 0 5 10 t t EE456 – Digital Communications 12
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. 0.5 2 Message m ( t ) s AM ( t ) 0 0 −0.5 −2 0 1 2 3 4 5 0 1 2 3 4 5 t t 1 2 s PM ( t ) d m ( t ) d t 0 0 −1 −2 0 1 2 3 4 5 0 1 2 3 4 5 t t 1 2 − ∞ m ( α )d α 0.5 s FM ( t ) 0 0 � t −0.5 −2 0 1 2 3 4 5 0 1 2 3 4 5 t t EE456 – Digital Communications 13
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 A 2 / 2 , regardless of the value of k p , k f , 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 � t a ( t ) = m ( α )d α −∞ A e j [ ω c t + k f a ( t )] = A e jk f a ( t ) e jω c t ⇒ s FM ( t ) ˆ = s FM ( t ) = ℜ{ ˆ s FM ( t ) } Expanding the exponential e jk f a ( t ) in power series gives: � � k 2 2! a 2 ( t ) + · · · + j n k n f f n ! a n ( t ) + · · · e jω c t s FM ( t ) ˆ = A 1 + jk f a ( t ) − s FM ( t ) = ℜ{ ˆ s FM ( t ) } � � k 2 k 3 f f 2! a 2 ( t ) cos( ω c t ) + 3! a 3 ( t ) sin( ω c t ) + · · · = A cos( ω c t ) − k f a ( t ) sin( ω c t ) − EE456 – Digital Communications 14
Chapter 3: Frequency Modulation (FM) Observations: The FM signal consists of an unmodulated carrier and various amplitude-modulated terms, such as a ( t ) sin( ω c t ) , a 2 ( t ) cos( ω c t ) , a 3 ( t ) sin( ω c t ) , 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 a 2 ( t ) is the spectrum of A ( f ) ∗ A ( f ) (where ∗ is the integral convolution operation) and is band-limited to [ − 2 B, 2 B ] . Similarly, the spectrum of a n ( t ) is band-limited to [ − nB, nB ] . The spectrum of s FM ( t ) consists of an unmodulated carrier, plus spectra of a ( t ) , a 2 ( t ) , . . . , a n ( t ) , . . . , centered at ω c . Clearly, the bandwidth of s FM ( t ) is theoretically infinite! For practical message signals, because n ! increases much faster than | k f a ( t ) | n , k n f a n ( t ) we have ≈ 0 for large n . Hence most of the modulated-signal power n ! resides in a finite bandwidth. Carson’s rule for Bandwidth Approximation of an FM Signal (captures 98% of total power): B FM = 2(∆ f + B ) = 2 B ( β + 1) m max − m min where ∆ f = k f is defined as the peak frequency deviation 2 · 2 π ∆ f β = is the deviation ratio B EE456 – Digital Communications 15
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