Properties of Angle Modulation

Properties of Angle Modulation Properties of Angle Modulation Reference: Sections 4.2 and 4.3 of Professor Deepa Kundur S. Haykin and M. Moher, Introduction to Analog & Digital Communications, 2nd ed., John Wiley & Sons, Inc., 2007. ISBN-13 978-0-471-43222-7. University of Toronto Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 1 / 25 Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 2 / 25 Angle Modulation carrier IPhase Modulation (PM): <ul style="display: flex;"><li style="flex:1">θi (t) </li><li style="flex:1">=</li><li style="flex:1">2πfc t + kpm(t) </li></ul>1 dθi (t) kp dm(t) message <ul style="display: flex;"><li style="flex:1">fi (t) </li><li style="flex:1">=</li></ul>= = fc + 2π dt Ac cos[2πfc t + kpm(t)] 2π dt sPM (t) amplitude modulation IFrequency Modulation (FM): Ztθi (t) fi (t) === 2πfc t + 2πkf m(τ)dτ 0phase modulation 1 dθi (t) = fc + kf m(t) 2π dt <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul>ZtsFM (t) Ac cos 2πfc t + 2πkf m(τ)dτ frequency modulation 0Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 3 / 25 Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 4 / 25 carrier carrier message message amplitude modulation amplitude modulation phase modulation phase modulation frequency modulation frequency modulation Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 5 / 25 Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 6 / 25 <ul style="display: flex;"><li style="flex:1">4.2 Properties of Angle Modulation </li><li style="flex:1">4.2 Properties of Angle Modulation </li></ul><ul style="display: flex;"><li style="flex:1">Constancy of Transmitted Power </li><li style="flex:1">Constancy of Transmitted Power: AM </li></ul>1f0 IConsider a sinusoid g(t) = Ac cos(2πf0t + φ) where T0 = or T0f0 = 1. IThe power of g(t) (over a 1 ohm resister) is defined as: ZZ ZT/2 T/2 <ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">1</li></ul>P===g2(t)dt = A2c cos2(2πf0t + φ)dt <ul style="display: flex;"><li style="flex:1">T0 </li><li style="flex:1">T0 </li></ul>−T /2 T −T /2 <ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul>/2 Ac2 T0 10[1 + cos(4πf0t + 2φ)] dt 2−T /2 0ꢂꢂꢂꢂ<ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul>T/2 Ac2 sin(4πf0t + 2φ) 0t + 2T0 4f0 −T /2 0<ul style="display: flex;"><li style="flex:1">ꢀꢃ </li><li style="flex:1">ꢄ</li><li style="flex:1">ꢃ</li><li style="flex:1">ꢄꢁ </li></ul>Ac2 2T0 Ac2 <ul style="display: flex;"><li style="flex:1">T0 </li><li style="flex:1">−T0 </li></ul>sin(4πf0T0/2 + 2φ) − sin(−4πf0T0/2 + 2φ) ==−+<ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">2</li></ul>4πf0 <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul>sin(2π + 2φ) − sin(−2π + 2φ) Ac2 T0 <ul style="display: flex;"><li style="flex:1">+</li><li style="flex:1">=</li></ul>2T0 4πf0 2ITherefore, the power of a sinusoid is NOT dependent on f0, just its envelop Ac . Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 7 / 25 Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 8 / 25 <ul style="display: flex;"><li style="flex:1">4.2 Properties of Angle Modulation </li><li style="flex:1">4.2 Properties of Angle Modulation </li></ul><ul style="display: flex;"><li style="flex:1">Constancy of Transmitted Power: PM </li><li style="flex:1">Constancy of Transmitted Power: FM </li></ul>Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 9 / 25 Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 10 / 25 <ul style="display: flex;"><li style="flex:1">4.2 Properties of Angle Modulation </li><li style="flex:1">4.2 Properties of Angle Modulation </li></ul><ul style="display: flex;"><li style="flex:1">Constancy of Transmitted Power </li><li style="flex:1">Nonlinearity of Angle Modulation </li></ul>Consider PM (proof also holds for FM). ISuppose s1(t) == Ac cos [2πfc t + kpm1(t)] s2(t) Ac cos [2πfc t + kpm2(t)] Therefore, angle modulated signals exhibit constancy of transmitted power. ILet m3(t) = m1(t) + m2(t). s3(t) == Ac cos [2πfc t + kp(m1(t) + m2(t))] s1(t) + s2(t) ∵ cos(2πfc t + A + B) = cos(2πfc t + A) + cos(2πfc t + B) Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 11 / 25 Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 12 / 25 <ul style="display: flex;"><li style="flex:1">4.2 Properties of Angle Modulation </li><li style="flex:1">4.2 Properties of Angle Modulation </li></ul><ul style="display: flex;"><li style="flex:1">Nonlinearity of Angle Modulation </li><li style="flex:1">Irregularity of Zero-Crossings </li></ul>IZero-crossing: instants of time at which waveform changes amplitude from positive to negative or vice versa. Therefore, angle modulation is nonlinear. Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 13 / 25 Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 14 / 25 <ul style="display: flex;"><li style="flex:1">4.2 Properties of Angle Modulation </li><li style="flex:1">4.2 Properties of Angle Modulation </li></ul><ul style="display: flex;"><li style="flex:1">Zero-Crossings: AM </li><li style="flex:1">Zero-Crossings: PM </li></ul>Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 15 / 25 Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 16 / 25 <ul style="display: flex;"><li style="flex:1">4.2 Properties of Angle Modulation </li><li style="flex:1">4.2 Properties of Angle Modulation </li></ul><ul style="display: flex;"><li style="flex:1">Zero-Crossings: FM </li><li style="flex:1">Irregularity of Zero-Crossings </li></ul>Therefore angle modulated signals exhibit irregular zero-crossings because they contain information about the message (which is irregular in general). Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 17 / 25 Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 18 / 25 <ul style="display: flex;"><li style="flex:1">4.2 Properties of Angle Modulation </li><li style="flex:1">4.2 Properties of Angle Modulation </li></ul><ul style="display: flex;"><li style="flex:1">Visualization Difficulty of Message </li><li style="flex:1">Visualization: AM </li></ul>IVisualization of a message refers to the ability to glean insights about the shape of m(t) from the modulated signal s(t). Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 19 / 25 Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 20 / 25 <ul style="display: flex;"><li style="flex:1">4.2 Properties of Angle Modulation </li><li style="flex:1">4.2 Properties of Angle Modulation </li></ul><ul style="display: flex;"><li style="flex:1">Visualization: PM </li><li style="flex:1">Visualization: FM </li></ul>Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 21 / 25 Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 22 / 25 <ul style="display: flex;"><li style="flex:1">4.2 Properties of Angle Modulation </li><li style="flex:1">4.2 Properties of Angle Modulation </li></ul><ul style="display: flex;"><li style="flex:1">Visualization Difficulty of Message </li><li style="flex:1">Bandwidth vs. Noise Trade-Off </li></ul>INoise affects the message signal piggy-backed as amplitude modulation more than it does when piggy-backed as angle modulation. Therefore it is difficult to visualize the message in angle modulated signals due to the nonlinear nature of the modulation process. IThe more bandwidth that the angle modulated signal takes, typically the more robust it is to noise. Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 23 / 25 Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 24 / 25 4.2 Properties of Angle Modulation carrier message amplitude modulation phase modulation frequency modulation Professor Deepa Kundur (University of Toronto) Properties of Angle Modulation 25 / 25 ꢀ

Properties of Angle Modulation

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