Chapter 3: Frequency Modulation (FM) EE456 – Digital Communications Professor Ha Nguyen September 2016 EE456 – Digital Communications 1
Chapter 3: Frequency Modulation (FM) ANALOG MODULATION Amplitude Modulation (AM) & Frequency Modulation (FM) EE456 – Digital Communications 2
Chapter 3: Frequency Modulation (FM) The Concept of Modulation Modulation refers to a process that puts the message signal into a specific frequency band in order to match the transmission characteristics of the physical channel (e.g. telephone channel, wireless LAN channel, etc.) Modulation can be classified into baseband and passband . The term “baseband” refers to the frequency band of the original message signal, which is usually near the zero frequency. For example, the band of audio (voice) signals is between 0 to 3.5kHz, the video baseband occupies 0 to 4.3 MHz. In baseband modulation, the message signals are directly transmitted over the channels (e.g. twisted pairs of copper wires, coaxial cables). Passband modulation is also known as carrier modulation, in which the spectrum of the message signal is shifted to a higher-frequency band by means of a sinusoidal carrier. Two important advantages of carrier modulation are: (i) To ease radio-frequency (RF) transmission, and (ii) to allow frequency division multiplexing. Carrier modulation can be analog or digital . Traditional communications such as AM/FM radios are based on analog modulation, while many current and new communications systems are all digital (cellular phone systems, HDTV, etc.) EE456 – Digital Communications 3
Chapter 3: Frequency Modulation (FM) Double-Sideband Suppressed-Carrier (DSB-SC) Amplitude Modulation Let m ( t ) be the message signal, whose Fourier transform is denoted as M ( f ) . To move the frequency band of m ( t ) to a new frequency band centered at f c , the message signal m ( t ) is simply multiplied by a sinusoid of frequency f c (i.e., the carrier): s DSB − SC ( t ) = m ( t ) cos(2 πf c t ) . The above signal is seen as a sinusoid of frequency f c whose amplitude is varied according to the message signal, hence the name amplitude modulation (AM). The spectrum of the AM signal s ( t ) is obtained as a convolution between the spectrum of m ( t ) and the spectrum of cos (2 πf c t ) . Since the spectrum of cos (2 πf c t ) is 1 2 δ ( f + f c ) + 1 2 δ ( f − f c ) , one has � 1 � 2 δ ( f + f c ) + 1 S DSB − SC ( f ) = M ( f ) ∗ 2 δ ( f − f c ) 1 2 M ( f − f c ) + 1 = 2 M ( f + f c ) . EE456 – Digital Communications 4
Chapter 3: Frequency Modulation (FM) Graphical Illustration of DSB-SC Amplitude Modulation m t ( ) = × π m t ( ) s DSB-SC ( ) t m t ( ) cos(2 f t ) c m t ( ) = π t c t ( ) cos(2 f t ) c t π m t ( )cos(2 f t ) M f ( ) c − m t ( ) S DSB-SC ( ) f C f ( ) 2 B 1 1 2 2 K K 2 2 f f f − − − − + B 0 B 0 0 f f f f B f f B c c c c c c The process of amplitude modulation shifts the spectrum of the modulating signal to the left and to the right by f c . If the bandwidth of m ( t ) is B Hz, then the modulated AM signal has bandwidth of 2 B Hz. There are upper sideband (USB) and lower sideband (LSB) portions of the AM spectrum. The modulated signal does not contain a discrete component of the carrier frequency f c . For this reason, the modulated signal in this scheme is double-sideband, suppressed-carrier (DSB-SC) modulation . EE456 – Digital Communications 5
Chapter 3: Frequency Modulation (FM) Coherent (Synchronous) Demodulation of DSB-SC AM Signals m t ( ) e t ( ) 2 m t ( ) = r t ( ) s ( ) t DSB-SC = π 2 c t ( ) cos(2 f t ) c E f ( ) K K K K 2 2 4 4 f f − − − − B B B 2 f f 0 f 2 f 0 B c c c c m ( t ) cos 2 (2 πf c t ) = 1 2 m ( t ) + 1 e ( t ) = 2 m ( t ) cos(2 π (2 f c ) t ) 1 2 M ( f ) + 1 E ( f ) = 4[ M ( f + 2 f c ) + M ( f − 2 f c )] This method of AM demodulation is called synchronous detection , or coherent detection , where the receiver requires to have a carrier of exactly the same frequency (and phase) as the carrier used for modulation. In practice, such a requirement is typically fulfilled with a phase-locked loop (PLL) circuit. EE456 – Digital Communications 6
Chapter 3: Frequency Modulation (FM) Example 1 Consider a single tone baseband signal m ( t ) = cos( ω m t ) = cos(2 πf m t ) . Find the DSB-SC signal and sketch its spectrum. Identify the USB and LSB. Also verify that the coherent demodulation works and recovers m ( t ) . (Partial) Solution: EE456 – Digital Communications 7
Chapter 3: Frequency Modulation (FM) Effect of Using a Non-Coherent Carrier at the Receiver = × π s DSB-SC ( ) t m t ( ) cos(2 f t ) c = − × π − r t ( ) m t ( t ) cos(2 f t ( t )) m t ( ) 0 c 0 1 − π ∆ + θ m t ( t )cos(2 ft ) = π 0 d e t ( ) c t f t 2 ( ) cos(2 ) c π + ∆ cos(2 ( f f t ) ) c To see the effect on the demodulated signal when the local oscillator at the receiver is not synchronized in frequency and phase with the incoming carrier, consider the case that the received signal at the receiver is a delayed version of the transmitted AM signal (due to propagation time): r ( t ) = m ( t − t 0 ) cos[2 πf c ( t − t 0 )] = m ( t − t 0 ) cos(2 πf c t − θ d ) , where θ d = 2 πf c t 0 is the equivalent phase shift. Furthermore, assume that, due to the lack of a good PLL circuit, the oscillator in the receiver generates cos(2 π ( f c + ∆ f ) t ) , i.e., there is a frequency offset of ∆ f compared to the carrier generated by the oscillator at the transmitter. EE456 – Digital Communications 8
Chapter 3: Frequency Modulation (FM) Then the signal e ( t ) is e ( t ) = m ( t − t 0 ) cos(2 πf c t − θ d ) cos(2 π ( f c + ∆ f ) t ) 1 2 m ( t − t 0 ) cos(2 π ∆ ft + θ d ) + 1 = 2 m ( t − t 0 ) cos(2 π (2 f c + ∆ f ) t − θ d ) The first component of e ( t ) is the message signal modulated with a very small “carrier” frequency of ∆ f , while the second component is the message signal modulated with a high carrier frequency of 2 f c + ∆ f . Thus, the output of the LPF will be the first component only, i.e., v ( t ) = 1 2 m ( t − t 0 ) cos(2 π ∆ ft + θ d ) . In the frequency domain, it can be shown that the spectrum of v ( t ) is V ( f ) = 1 4 e jθ d M ( f − ∆ f )e − j 2 π ( f − ∆ f ) t 0 + 1 4e − jθ d M ( f + ∆ f )e − j 2 π ( f +∆ f ) t 0 In essence, the spectrum at the output of the LPF is the spectrum of the original message moved to ± ∆ f . In Lab #1 you will hear the effect of this frequency translation for different values of ∆ f when m ( t ) is an audio signal. EE456 – Digital Communications 9
Chapter 3: Frequency Modulation (FM) Amplitude Modulation with Carrier + m t ( ) m t ( ) A t t 0 + δ M ( f ) A ( f ) C R f − B 0 B ∑ m t ( ) = + × π m t ( ) s DSB ( ) t [ ( ) m t A ] cos(2 f t ) c = π c t ( ) cos(2 f t ) t c t t M f ( ) S f C f ( ) DSB ( ) 2 B 1 1 2 2 A A K K 2 2 2 2 f f f − − − − + B 0 B f 0 f f 0 f B f f B c c c c c c EE456 – Digital Communications 10
Chapter 3: Frequency Modulation (FM) To enable a simpler receiver than the coherent receiver, the transmitter can send a carrier along with the modulated signal: s AM ( t ) = A cos(2 πf c t )+ m ( t ) cos(2 πf c t ) = [ A + m ( t )] cos(2 πf c t ) , where A > 0 . The spectrum of s AM ( t ) is basically the same as that of s DSB − SC ( t ) = m ( t ) cos(2 πf c t ) , except for two additional impulses at ± f c : S AM ( f ) = 1 2 M ( f − f c ) + 1 2 M ( f + f c ) + A 2 δ ( f − f c ) + A 2 δ ( f + f c ) . If the carrier component is large enough, the message signal m ( t ) can be recovered with a very simple envelope detector (see the next slides). The option of AM with carrier is very desirable in broadcasting systems as it offers a trade-off to have one expensive high-power transmitter and (many) inexpensive receivers. EE456 – Digital Communications 11
Chapter 3: Frequency Modulation (FM) Envelope Detection of AM Signals The time constant of the RC circuit should be such that the capacitor discharges very little between positive peaks and in a similar rate of the AM envelope variation: 1 1 /ω c ≪ RC < 1 / (2 πB ) , or 2 πB < RC ≪ ω c , where B is the bandwidth of m ( t ) . The envelope detector output is A + m ( t ) with a ripple frequency ω c . The DC term can be blocked by a simple highpass filter, while the ripple may be further reduced by another lowpass filter. EE456 – Digital Communications 12
Chapter 3: Frequency Modulation (FM) Envelope of an AM Signal By definition, the envelope of the AM signal is | A + m ( t ) | . If A is large enough to ensure that A + m ( t ) ≥ 0 for all t , then the envelope has the same shape as the message m ( t ) . This means that we can detect the desired signal m ( t ) by detecting the envelope of the AM signal! If A + m ( t ) < 0 for some time t , then the envelope | A + m ( t ) | does not have the same shape as the message m ( t ) and envelope detection does not work correctly. EE456 – Digital Communications 13
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