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First semester 1435-14356 King Saud University College of Applied studies and Community Service 1301CT

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Outline Sampling Digital communication system Analog pulse modulation Digital pulse modulation

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Sampling Process The sampling process is a basic operation in the digital communication. In this process, the continuous-time analog signal signal is sampled by measuring its amplitude at a discrete instants. So, the continuous-time analog signal is converted into a corresponding sequence of samples that are usually spaced uniformly in time. It is necessary to choose the sampling rate properly, so the sequence of samples uniquely defines the original analog signal.

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The Sampling Theorem The sampling theorem shows that a continuous-time signal which is a band-limited to f m Hz can be represented perfectly by a series of samples spaced Ts [ ≤ 1/(2 f m )] seconds apart. Ts called sampling period The band-limited signal is a signal which its Fourier transform is nonzero for -2 π f m < ω < 2 π f m. When m(t) is sampled uniformly at intervals of Ts seconds, the resultant sequence is denoted by m(nTs), for all integer values of n.

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S(t) is band- limited signal Ts sampling period

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The Sampling Theorem Let’s define a sampling rate fs = 1 / Ts which is the number of samples per second. Thus, the sampling theorem states that a band-limited signal can be recovered completely from a set of samples taken at the rate of fs ( ≥ 2 f m ) samples per second. The preceding sampling theorem is often called the uniform sampling theorem for baseband or low-pass signal. The minimum sampling rate, 2f m samples per second, called the Nyquist rate; its reciprocal 1/2f m is called Nyquist interval

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Sampling Process Sampling g(t) at a rate of f s Hz (f s samples per second) can be accomplished by multiplying g(t) by an impulse train δ Ts (t), consisting of unit impulses repeating periodically every T s seconds, where T s = 1 / f s Let g(t) be a signal whose spectrum is band-limited to f m Hz

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Sampling Process This results sampled signal ğ(t) consists of impulses spaced every T s seconds. The n th impulse, located at t=n T s has a strength g(nT s ), the value of g(t) at t = n T s

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Sampling Process

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2 f m 1/2f m

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Aliasing What happens if we sample the signal at a frequency that is lower that the Nyquist rate? If the Nyquist criterion is not satisfied, the adjacent copies of g(t) spectrum will overlap, this phenomenon called aliasing

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Aliasing With aliasing : - Some of the frequencies in the original signal will be lost in the reconstructed signal - unwanted components will be presence in the reconstructed signal. these components were not present when the original signal was sampled. In addition,. Aliasing occurs because signal frequencies will overlap if the sampling frequency is too low.

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Example What Nyquist rate is needed for a signal with a bandwidth 10,000Hz ( 1000 to 11,000 Hz)? The Nyquist rate is equal to twice the highest frequency in the signal Nyquist rate = 2 * 11,000 = 22,000 samples/sec

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