EE456 Digital Communications

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1 EE456 Digital Communications Professor Ha Nguyen September 216 EE456 Digital Communications 1

2 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

3 Instantaneous Frequency Consider a generalized sinusoidal signal c(t) = Acosθ(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 ct+θ. It is clear from the figure that, over the interval t 1 < t < t 2 one has: c(t) = Acosθ(t) = Acos(ω ct+θ ), 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

4 For a conventional sinusoid Acos(ω ct+θ ), the generalized angle is a straight line ω ct+θ 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) = dθ(t) dt The equivalent relationship between angle θ(t) and the instantaneous frequency ω i (t) is: t θ(t) = ω i (α)dα EE456 Digital Communications 4

5 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 ct+k pm(t), (assuming θ = ) s PM (t) = Acos[ω ct+k pm(t)], (where k p is a constant) The instantaneous angular frequency ω i (t) of the PM signal is ω i (t) = dθ(t) dt dm(t) = ω c +k p, dt 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)dα = [ω c +k f m(α)]dα = ω ct+k f m(α)dα t ] s FM (t) = Acos [ω ct+k f m(α)dα EE456 Digital Communications 5

6 Relationship Between FM and PM s () t FM s () t PM 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), dm(t), or m(α)dα can dt be treated as a message signal. EE456 Digital Communications 6

7 PM and FM Circuits (Analog) Note: RFC stands for radio-frequency choke EE456 Digital Communications 7

8 Example 3 The figure below shows a message signal m(t) and its derivative. Suppose that the constants k f and k p are 2π 1 5 and 1π, respectively, and the carrier frequency f c is 1 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

9 Solution: (a) For FM, we have: f i (t) = ω i(t) 2π = fc + k f 2π m(t) = m(t) [f i (t)] min = [m(t)] min = 99.9 MHz [f i (t)] max = [m(t)] max = 1.1 MHz (b) For PM, we have: f i (t) = ω i(t) 2π = fc + kp 2πṁ(t) = 18 +5ṁ(t) [f i (t)] min = [ṁ(t)] min = 99.9 MHz [f i (t)] max = [ṁ(t)] max = 1.1 MHz EE456 Digital Communications 9

10 (c) Sketches of the FM and PM signals are shown below. s () t FM s () t PM Observations: Because m(t) increases and decreases linearly with time, the instantaneous frequency of the FM signal increases linearly from 99.9 to 1.1 MHz over a half-cycle, and then decreases linearly from 1.1 MHz to 99.9 MHz over the remaining half-cycle. Because ṁ(t) switches back and forth from a value of 2, to 2,, the carrier frequency switches back and forth from 99.9 to 1.1 MHz every half-cycle of ṁ(t). EE456 Digital Communications 1

11 Comparison of AM, FM and PM Signals with the same massage m(t) Message m(t) Can you tell which signals on the right are AM, FM and PM, respectively? t t 2 dm(t) dt t m(α)dα t t t t EE456 Digital Communications 11

12 Comparison of AM, FM and PM Signals with the same massage m(t) Message m(t) t 1 sam(t) t 2 dm(t) dt spm(t) t m(α)dα t t sfm(t) t t EE456 Digital Communications 12

13 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. Message m(t) t 1 sam(t) t 2 dm(t) dt spm(t) t m(α)dα t t sfm(t) t t EE456 Digital Communications 13

14 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 a(t) = t m(α)dα ŝ FM (t) = Ae j[ωct+k fa(t)] = Ae jk fa(t) e jωct s FM (t) = R{ŝ FM (t)} Expanding the exponential e jkfa(t) in power series gives: [ ŝ FM (t) = A s FM (t) = R{ŝ FM (t)} [ = A 1+jk f a(t) k2 f 2! a2 (t)+ +j nkn f n! an (t)+ ] e jωct cos(ω ct) k f a(t)sin(ω ct) k2 f 2! a2 (t)cos(ω ct)+ k3 f 3! a3 (t)sin(ω ct)+ ] EE456 Digital Communications 14

15 Observations: The FM signal consists of an unmodulated carrier and various amplitude-modulated terms, such as a(t)sin(ω ct), a 2 (t)cos(ω ct), a 3 (t)sin(ω ct), 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 [ 2B, 2B]. 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, we have kn f an (t) 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) = 2B(β +1) m max m min where f = k f is defined as the peak frequency deviation 2 2π β = f is the deviation ratio B EE456 Digital Communications 15

16 Spectral Analysis of Tone FM When the message m(t) is a sinusoid, namely m(t) = A mcos(ω mt), and with the initial condition a( ) =, one has a(t) = Am ω m sin(ω mt) β = f B = Amk f ω m ŝ FM (t) = Ae (jωct+jk fa m/ω m sin(ω mt)) = Ae (jωct+jβsin(ωmt)) = Ae jωct( jβ e sin(ωmt)) Since e jβ sin(ωmt) is a periodic signal with period T = 2π/ω m, it can be expanded by the exponential Fourier series: e jβ sin(ωmt) = where D n D ne jnωmt n= = ωm π/wm e jβ sin(ωmt) e jnωmt dt 2π π/ω m 1 π = e j(β sinx nx) dx = J n(β) 2π π }{{} nth-order Bessel function of the first kind EE456 Digital Communications 16

17 It then follows that Observations: ŝ FM (t) = A J n(β)e j(ωct+nωmt) n= s FM (t) = A J n(β)cos((ω c +nω m)t) n= The tone-modulated FM signal has a carrier component and an infinite number of sidebands of frequencies ω c ±ω m, ω c ±2ω m,...,ω c ±nω m. This is very different from DSB-SC spectrum of tone-modulated AM signal! The strength of the nth sideband at ω c +nω m is A Jn(β), which quickly 2 decreases with n. In fact, there are only a finite number of significant sideband spectral lines. In general, J n(β) is negligible for n > β +1, hence the bandwidth of tone-modulated FM signal is approximated as: B FM = 2(β +1)f m = 2( f +B) EE456 Digital Communications 17

18 Plot of J n (β) J (β) Jn(β) J 1 (β) J 2 (β) J 3 (β) J 4 (β) J 5 (β) J 6 (β) Two important properties: J n (β) = ( 1) n J n(β) β n= J 2 n (β) = 1 EE456 Digital Communications 18

19 Table of J n (β) EE456 Digital Communications 19

20 Illustration of Tone FM Spectrum S ( ) FM f A /2 S ( ) FM f A /2 S ( ) FM f A /2 S ( ) FM f A /2 S ( ) FM f A /2 EE456 Digital Communications 2

Example 4 The figure below shows a message signal m(t) and its derivative. Suppose that the constant k f = 2π 1 5.

21 Example 4 The figure below shows a message signal m(t) and its derivative. Suppose that the constant k f = 2π (a) Since m(t) is periodic with a fundamental frequency f = 2 1 4, it can be represented as m(t) = k= a ke j2πkft. Show that a = and { 4 a k = π 2 k 2, k odd, k even (b) Assume that the essential bandwidth of m(t) to be the frequency of its third harmonic, estimate the bandwidth of the FM signal when the modulating signal is m(t). (c) Repeat Part (b) if the amplitude of m(t) is doubled (i.e., if m(t) is multiplied by 2). (d) Repeat Part (b) if m(t) is time-expanded by a factor of 2 (i.e., if the period of m(t) is ). EE456 Digital Communications 21

22 Narrow-Band FM (NBFM) s FM (t) = A [ ] cos(ω ct) k f a(t)sin(ω k2 f ct) 2! a2 (t)cos(ω k3 f ct)+ 3! a3 (t)sin(ω ct)+ When k f is very small such that k f a(t) 1, then all higher order terms in the above expression are negligible, except for the first two terms. We then have a good approximation of an FM signal: s FM (t) A [ cos(ω ct) k f a(t)sin(ω ct) ] (1) The above approximation is a linear modulation similar to that of an AM signal with the message signal being a(t). Because the bandwidth of a(t) is the same as the bandwidth of m(t), which is B Hz, the bandwidth of the narrowband FM signal in (1) is 2B Hz. It is pointed out that the sideband spectrum for a NBFM signal has a phase shift of π/2 with respect to the carrier, whereas the sideband spectrum of an AM signal is in phase with the carrier. The expression of the NBFM signal in (1) suggests a method of generating a NBFM signal by using a DSB-SC modulator (see Fig. 1-(a) on the next slide). The output of the NBFM modulator in Fig. 1-(a) has some amplitude variations (distortion). Such distortion can be removed by using a hard-limiter and a bandpass filter as shown in Fig. 1-(b). The analysis of Fig. 1-(b) shall be explored in Assignment 2. EE456 Digital Communications 22

23 m( t) a( t) Asin( ω t) Ak a( t)sin( ω t) c f c NBFM signal A( t)cos[ ω t + ϕ( t)] c π 2 Acos( ω t) c A( t)cos[ ω t + ϕ( t)] c 4 cos[ ω t c + ϕ ( t )] π Figure 1: Generating a NBFM signal. EE456 Digital Communications 23

24 Demodulation of FM Signals ] Signal at point b : s FM (t) = Acos [ω t ct+k f m(α)dα Signal at point c : { t ]} Acos [ω ct+k f m(α)dα ṡ FM (t) = d dt t = A[ω c +k f m(t)]sin [ω ct+k f Signal at point d : A[ω c +k f m(t)] Signal at point e : k f m(t) m(α)dα π ] EE456 Digital Communications 24

25 A Practical (Continuous-Time) Differentiator Recall that the frequency response of an ideal differentiator is H(f) = j2πf. A differentiator can be approximated by a linear system whose frequency response contains a linear segment of a positive slope. One simple device would be an RC high-pass filter. The RC frequency response is simply H(f) = j2πfrc j2πfrc, if 2πfRC 1. 1+j2πfRC Thus, if the parameter RC is very small such that its product with the carrier frequency ω crc 1, the RC filter approximates a differentiator. EE456 Digital Communications 25

26 FCC FM Standards EE456 Digital Communications 26

27 FM Stations in Saskatoon EE456 Digital Communications 27

28 Stereo FM EE456 Digital Communications 28

29 Review of Discrete-Time Processing of Continuous-Time Signals H ( ) c jω H e ω jˆ d( ) The frequency response of the discrete-time LTI system, H d (e jŵ ) is periodic with period 2π. Over π ŵ π, it is simply a frequency-scaled version of H c(ω): H d (e jŵ ) = H c(ˆωf s), π ŵ π where f s = 1 T is the sampling frequency. EE456 Digital Communications 29

30 Example: Discrete-Time Low-Pass Filter H ( ) c jω H ( ) c jω H e ω jˆ d( ) H e ω jˆ d( ) ˆω EE456 Digital Communications 3

31 Discrete-Time Integrator EE456 Digital Communications 31

32 An illustration of the backward difference, forward difference, and trapezoid rule for approximating the integral of a continuous-time signal using discrete-time processing. EE456 Digital Communications 32

33 Realization of discrete-time integrators: (a) A realization of the discrete-time integrator based on the trapezoid rule. (b) A realization of the discrete-time integrator based on the backward difference. (c) A rearrangement of (b) to produce the more traditional system block diagram of an accumulator. EE456 Digital Communications 33

34 Freq. Responses: Ideal Integrator, Accumulator, Trapezoid-Rule Integrator Ideal Integrator: H ideal (e jŵ ) = 1 jŵ. Accumulator: H acc(z) = 1 1 z 1, H acc(e jŵ ) = 1 1 e jŵ. Trapezoid-Rule Integrator: H trap(z) =.5 1+z 1 1 z 1, H trap(e jŵ ) =.5 1+e jŵ 1 e jŵ Hideal(e jˆω ) Hacc(e jˆω ) Htrap(e jˆω ) Magnitude response π 3 2 π/2 1 1 π/2 2 3 π ˆω (radians/sample) The accumulator works very well as a DT integrator, especially for small-bandwidth signals. EE456 Digital Communications 34

35 Discrete-Time Differentiator H e ω jˆ d( ) H e ω jˆ d( ) ˆω EE456 Digital Communications 35

36 H(ω) = h d [n] = { jω, ω Wc, otherwise 1 WcT 2π W ct { H d (e jŵ j ˆω, ˆω WcT ) = T, W ct < ˆω π j ˆω T ejŵn dˆω = WcT πt cos(w ctn) n 1 sin(w ctn) πt n 2 The impulse response has infinite support The discrete-time system is an IIR filter. For the special case of full-bandwidth, i.e., when W ct = π, the impulse response is { 1 ( 1) n h d [n] = T n, n, n = The first few samples of the impulse response for the full-bandwidth differentiator are shown below. EE456 Digital Communications 36

37 An Approximate Discrete-Time Differentiator By truncating the impulse response to n = 1,,1, the differentiator consists of the three center coefficients. The output of such a differentiator is y[n] = 1 T (x[n+1] x[n 1]) The above system is non-causal. It can be made causal by introducing a delay of 1 sample: y[n] = 1 T (x[n] x[n 2]) approximate a differentiator with a delay of 1 sample xn [ ] 1 z z 1 yn [ ] = xn [ ] xn [ 2] + Ignoring the scaling factor 1, the impulse response of the above approximate T differentiator is h[n] = δ[n] δ[n 2]. EE456 Digital Communications 37

38 The system function H(z) = 1 z 2 has 2 zeros at and π. The frequency response is H(e jŵ ) = 1 z z=e 2 = 1 e j2ŵ jŵ = e jŵ (e jŵ e jŵ ) = e } jŵ {{} (2j sin ˆω }{{} ) 2jˆωe jŵ delay of 1 sample ˆω for ˆω small Happrox(e jˆω ) Hideal(e jˆω ) Magnitude response π π/2 ˆω (radians/sample) π/2 π The above length-3 FIR approximation to a differentiator works reasonably well for small-bandwidth signal, about ˆω.2π EE456 Digital Communications 38

39 Better Approximations of a Discrete-Time Differentiator Use a Blackman window (Matlab command blackman) to approximate an ideal differentiator as an FIR filter Frequency response N=3, 7, 11, 15, 19, 23, 27, frequency (cycles/sample) EE456 Digital Communications 39

40 1 Frequency response (db) N=3, 7, 11, 15, 19, 23, 27, frequency (cycles/sample) EE456 Digital Communications 4

41 Building FM Transmitter and Receiver in Lab # 3 Transmitter m[ n] + FM s [ n] = cos( ω n + θ[ n]) c k f + θ[ n] = θ[ n 1] + 2 π k f m[ n] (cycles/sample) f c EE456 Digital Communications 41

42 Receiver s [ n] FM x [ ] c n cos[( ω + ω) n] c f + f c sin[( ω + ω) n] c x [ ] s n y [ ] c n y [ 1] c n y [ ] s n y [ 1] s n 1 z z 1 1 z z y ' c[ n 1] y ' s[ n 1] + θ ' [ n 1] EE456 Digital Communications 42

43 Analysis of the FM Demodulator t ] s FM (t) = cos [ω ct+k f m(α)dα = cos[ω ct+θ(t)] s FM [n] = cos[ω cnt s +θ(nt s)] = cos(ˆω cn+θ[n]) y c[n] = cos( ˆωn θ[n]); where ˆω = 2π ˆf y s[n] = sin( ˆωn θ[n]); y c [n 1] d dt yc(t), where y c(t) = cos( ωt θ(t)), ω = t=(n 1)Ts ˆω T s = ( ω θ (t))sin( ωt θ(t)) t=(n 1)Ts Similarly, = ( ˆω θ [n 1])sin( ˆω(n 1) θ[n 1]) y s [n 1] ( ˆω θ [n 1])cos( ˆω(n 1) θ[n 1]) Finally, y c[n 1]y s[n 1] y s[n 1]y c[n 1] = (θ [n 1] ˆω)[cos 2 ( ˆω(n 1) θ[n 1])+sin 2 ( ˆω(n 1) θ[n 1])] = (θ [n 1] ˆω) = θ [n 1] ˆω In the above ˆω is the DC offset due to error in the receiver s local oscillator, while θ [n 1] is proportional to m[n 1]. EE456 Digital Communications 43

ELE636 Communication Systems

ELE636 Communication Systems Chapter 5 : Angle (Exponential) Modulation 1 Phase-locked Loop (PLL) The PLL can be used to track the phase and the frequency of the carrier component of an incoming signal.