The plot of for the hypothetical case happens to be

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The plot of 𝜃 ? for the hypothetical case happens to be tangential to the angle 𝜔 𝑐 ? + 𝜃 0 at some instant t. Talha Asghar 7
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Instantaneous Frequency (cont.) The crucial point is that over a small interval Δ? → 0 , both signals are identical. Over this small interval Δ? the frequency of 𝜙 ? is 𝜔 𝑐 . Frequency is actually the slope of its angle over this interval. Talha Asghar 8
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Instantaneous Frequency (cont.) The instantaneous frequency is 𝜔 𝑖 ? = ?𝜃 ?? 𝜃 ? = න −∞ 𝑡 𝜔 𝑖 𝛼 ?𝛼 Now we can see the possibility of transmitting the information of m(t) by varying the angle of a carrier. Talha Asghar 9
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Instantaneous Frequency (cont.) Such techniques of modulation, where the angle of the carrier is varied in some manner with a modulating signal m(t), are known as angle modulation or exponential modulation. Two simple possibilities are Phase modulation (angle is varied linearly with m(t) ) Frequency modulation (instantaneous frequency is modified directly with the amplitude of m(t) ) Talha Asghar 10
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Phase Modulation In PM, the angle is varied linearly with m(t) 𝜃 ? = 𝜔 𝑐 ? + 𝜃 0 + 𝑘 𝑝 𝑚(?) Assuming the initial phase to be zero 𝜃 ? = 𝜔 𝑐 ? + 𝑘 𝑝 𝑚(?) The resulting PM wave is 𝜙 𝑃𝑀 ? = 𝐴 cos 𝜔 𝑐 ? + 𝑘 𝑝 𝑚(?) Talha Asghar 11
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Phase Modulation (cont.) The instantaneous frequency in this case is 𝜔 𝑖 ? = ?𝜃 ??
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  • Fall '19
  • Talha Asghar

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