
<p>Properties of Angle Modulation <br>Properties of Angle Modulation </p><p>Reference: </p><p>Sections 4.2 and 4.3 of <br>Professor Deepa Kundur </p><p>S. Haykin and M. Moher, Introduction to Analog & Digital Communications, 2nd ed., John Wiley & Sons, Inc., 2007. ISBN-13 978-0-471-43222-7. </p><p>University of Toronto </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>1 / 25 </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>2 / 25 </p><p>Angle Modulation </p><p>carrier </p><p>I</p><p>Phase Modulation (PM): </p><ul style="display: flex;"><li style="flex:1">θ<sub style="top: 0.1234em;">i </sub>(t) </li><li style="flex:1">=</li><li style="flex:1">2πf<sub style="top: 0.1234em;">c </sub>t + k<sub style="top: 0.1234em;">p</sub>m(t) </li></ul><p>1 dθ<sub style="top: 0.1233em;">i </sub>(t) </p><p>k<sub style="top: 0.1233em;">p </sub>dm(t) </p><p>message </p><p></p><ul style="display: flex;"><li style="flex:1">f<sub style="top: 0.1233em;">i </sub>(t) </li><li style="flex:1">=</li></ul><p>=<br>= f<sub style="top: 0.1233em;">c </sub>+ </p><p>2π dt </p><p>A<sub style="top: 0.1233em;">c </sub>cos[2πf<sub style="top: 0.1233em;">c </sub>t + k<sub style="top: 0.1233em;">p</sub>m(t)] </p><p>2π dt </p><p>s<sub style="top: 0.1233em;">PM </sub>(t) </p><p>amplitude modulation </p><p>I</p><p>Frequency Modulation (FM): </p><p>Z</p><p>t</p><p>θ<sub style="top: 0.1233em;">i </sub>(t) f<sub style="top: 0.1233em;">i </sub>(t) <br>===<br>2πf<sub style="top: 0.1233em;">c </sub>t + 2πk<sub style="top: 0.1233em;">f </sub></p><p>m(τ)dτ </p><p>0</p><p>phase modulation </p><p>1 dθ<sub style="top: 0.1233em;">i </sub>(t) <br>= f<sub style="top: 0.1233em;">c </sub>+ k<sub style="top: 0.1233em;">f </sub>m(t) </p><p>2π dt </p><p></p><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><p>Z</p><p>t</p><p>s<sub style="top: 0.1234em;">FM </sub>(t) </p><p>A<sub style="top: 0.1234em;">c </sub>cos 2πf<sub style="top: 0.1234em;">c </sub>t + 2πk<sub style="top: 0.1234em;">f </sub></p><p>m(τ)dτ </p><p>frequency modulation </p><p>0</p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>3 / 25 </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>4 / 25 </p><p>carrier </p><p>carrier </p><p>message </p><p>message </p><p>amplitude modulation </p><p>amplitude modulation </p><p>phase modulation </p><p>phase modulation </p><p>frequency modulation </p><p>frequency modulation </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>5 / 25 </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>6 / 25 </p><p></p><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><p></p><p></p><ul style="display: flex;"><li style="flex:1">Constancy of Transmitted Power </li><li style="flex:1">Constancy of Transmitted Power: AM </li></ul><p></p><p>1</p><p>f<sub style="top: 0.0822em;">0 </sub></p><p>I</p><p>Consider a sinusoid g(t) = A<sub style="top: 0.1233em;">c </sub>cos(2πf<sub style="top: 0.1233em;">0</sub>t + φ) where T<sub style="top: 0.1233em;">0 </sub>= or T<sub style="top: 0.1233em;">0</sub>f<sub style="top: 0.1233em;">0 </sub>= 1. </p><p>I</p><p>The power of g(t) (over a 1 ohm resister) is defined as: </p><p>ZZ<br>Z</p><p>T</p><p>/2 </p><p>T</p><p>/2 </p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">1</li></ul><p></p><p>P</p><p>===</p><p>g<sup style="top: -0.2732em;">2</sup>(t)dt = </p><p>A<sup style="top: -0.2732em;">2</sup><sub style="top: 0.1644em;">c </sub>cos<sup style="top: -0.2732em;">2</sup>(2πf<sub style="top: 0.0972em;">0</sub>t + φ)dt </p><p></p><ul style="display: flex;"><li style="flex:1">T<sub style="top: 0.0971em;">0 </sub></li><li style="flex:1">T<sub style="top: 0.0971em;">0 </sub></li></ul><p></p><p>−T /2 T<br>−T /2 </p><p></p><ul style="display: flex;"><li style="flex:1">0</li><li style="flex:1">0</li></ul><p></p><p>/2 </p><p>A<sub style="top: 0.1644em;">c</sub><sup style="top: -0.2321em;">2 </sup></p><p>T<sub style="top: 0.0972em;">0 </sub></p><p>1</p><p>0</p><p>[1 + cos(4πf<sub style="top: 0.0971em;">0</sub>t + 2φ)] dt <br>2</p><p>−T /2 </p><p>0</p><p>ꢂꢂꢂꢂ</p><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><p></p><p>T</p><p>/2 </p><p>A<sub style="top: 0.1644em;">c</sub><sup style="top: -0.232em;">2 </sup></p><p>sin(4πf<sub style="top: 0.0972em;">0</sub>t + 2φ) </p><p>0</p><p>t + </p><p>2T<sub style="top: 0.0971em;">0 </sub></p><p>4f<sub style="top: 0.0971em;">0 </sub></p><p>−T /2 </p><p>0</p><p></p><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><p></p><p>A<sub style="top: 0.1644em;">c</sub><sup style="top: -0.2321em;">2 </sup></p><p>2T<sub style="top: 0.0972em;">0 </sub></p><p>A<sub style="top: 0.1644em;">c</sub><sup style="top: -0.232em;">2 </sup></p><p></p><ul style="display: flex;"><li style="flex:1">T<sub style="top: 0.0972em;">0 </sub></li><li style="flex:1">−T<sub style="top: 0.0972em;">0 </sub></li></ul><p></p><p>sin(4πf<sub style="top: 0.0972em;">0</sub>T<sub style="top: 0.0972em;">0</sub>/2 + 2φ) − sin(−4πf<sub style="top: 0.0972em;">0</sub>T<sub style="top: 0.0972em;">0</sub>/2 + 2φ) <br>=</p><p>=</p><p>−</p><p>+</p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">2</li></ul><p></p><p>4πf<sub style="top: 0.0972em;">0 </sub></p><p></p><ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢁ</li></ul><p></p><p>sin(2π + 2φ) − sin(−2π + 2φ) </p><p>A<sub style="top: 0.1644em;">c</sub><sup style="top: -0.232em;">2 </sup></p><p>T<sub style="top: 0.0972em;">0 </sub></p><p></p><ul style="display: flex;"><li style="flex:1">+</li><li style="flex:1">=</li></ul><p></p><p>2T<sub style="top: 0.0971em;">0 </sub></p><p>4πf<sub style="top: 0.0971em;">0 </sub></p><p>2</p><p>I</p><p>Therefore, the power of a sinusoid is NOT dependent on f<sub style="top: 0.1233em;">0</sub>, just its envelop </p><p>A<sub style="top: 0.1233em;">c </sub>. </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>7 / 25 </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>8 / 25 </p><p></p><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><p></p><p></p><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><p></p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>9 / 25 </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>10 / 25 </p><p></p><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><p></p><p></p><ul style="display: flex;"><li style="flex:1">Constancy of Transmitted Power </li><li style="flex:1">Nonlinearity of Angle Modulation </li></ul><p></p><p>Consider PM (proof also holds for FM). </p><p>I</p><p>Suppose </p><p>s<sub style="top: 0.1233em;">1</sub>(t) </p><p>==<br>A<sub style="top: 0.1233em;">c </sub>cos [2πf<sub style="top: 0.1233em;">c </sub>t + k<sub style="top: 0.1233em;">p</sub>m<sub style="top: 0.1233em;">1</sub>(t)] </p><p>s<sub style="top: 0.1233em;">2</sub>(t) </p><p>A<sub style="top: 0.1233em;">c </sub>cos [2πf<sub style="top: 0.1233em;">c </sub>t + k<sub style="top: 0.1233em;">p</sub>m<sub style="top: 0.1233em;">2</sub>(t)] </p><p>Therefore, angle modulated signals exhibit constancy of transmitted </p><p>power. </p><p>I</p><p>Let m<sub style="top: 0.1233em;">3</sub>(t) = m<sub style="top: 0.1233em;">1</sub>(t) + m<sub style="top: 0.1233em;">2</sub>(t). </p><p>s<sub style="top: 0.1233em;">3</sub>(t) </p><p>==<br>A<sub style="top: 0.1233em;">c </sub>cos [2πf<sub style="top: 0.1233em;">c </sub>t + k<sub style="top: 0.1233em;">p</sub>(m<sub style="top: 0.1233em;">1</sub>(t) + m<sub style="top: 0.1233em;">2</sub>(t))] </p><p>s<sub style="top: 0.1233em;">1</sub>(t) + s<sub style="top: 0.1233em;">2</sub>(t) </p><p>∵ cos(2πf<sub style="top: 0.1233em;">c </sub>t + A + B) = cos(2πf<sub style="top: 0.1233em;">c </sub>t + A) + cos(2πf<sub style="top: 0.1233em;">c </sub>t + B) </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>11 / 25 </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>12 / 25 </p><p></p><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><p></p><p></p><ul style="display: flex;"><li style="flex:1">Nonlinearity of Angle Modulation </li><li style="flex:1">Irregularity of Zero-Crossings </li></ul><p></p><p>I</p><p>Zero-crossing: instants of time at which waveform changes amplitude from positive to negative or vice versa. <br>Therefore, angle modulation is nonlinear. </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>13 / 25 </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>14 / 25 </p><p></p><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><p></p><p></p><ul style="display: flex;"><li style="flex:1">Zero-Crossings: AM </li><li style="flex:1">Zero-Crossings: PM </li></ul><p></p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>15 / 25 </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>16 / 25 </p><p></p><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><p></p><p></p><ul style="display: flex;"><li style="flex:1">Zero-Crossings: FM </li><li style="flex:1">Irregularity of Zero-Crossings </li></ul><p></p><p>Therefore angle modulated signals exhibit irregular zero-crossings </p><p>because they contain information about the message (which is irregular in general). </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>17 / 25 </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>18 / 25 </p><p></p><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><p></p><p></p><ul style="display: flex;"><li style="flex:1">Visualization Difficulty of Message </li><li style="flex:1">Visualization: AM </li></ul><p></p><p>I</p><p>Visualization of a message refers to the ability to glean insights about the shape of m(t) from the modulated signal s(t). </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>19 / 25 </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>20 / 25 </p><p></p><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><p></p><p></p><ul style="display: flex;"><li style="flex:1">Visualization: PM </li><li style="flex:1">Visualization: FM </li></ul><p></p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>21 / 25 </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>22 / 25 </p><p></p><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><p></p><p></p><ul style="display: flex;"><li style="flex:1">Visualization Difficulty of Message </li><li style="flex:1">Bandwidth vs. Noise Trade-Off </li></ul><p></p><p>I</p><p>Noise affects the message signal piggy-backed as amplitude modulation more than it does when piggy-backed as angle </p><p>modulation. </p><p>Therefore it is difficult to visualize the message in angle modulated </p><p>signals due to the nonlinear nature of the modulation process. </p><p>I</p><p>The more bandwidth that the angle modulated signal takes, typically the more robust it is to noise. </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>23 / 25 </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>24 / 25 </p><p>4.2 Properties of Angle Modulation </p><p>carrier message amplitude modulation </p><p>phase modulation </p><p>frequency modulation </p><p>Professor Deepa Kundur (University of Toronto) </p><p>Properties of Angle Modulation </p><p>25 / 25 </p><p>ꢀ</p>
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