1. Qassim University College of Engineering Electrical Engineering Department Course: EE301: Signals and Systems Analysis The sampling Process Instructor: Associate Prof. Dr. Ahmed Abdelwahab

2. The sampling process The sampling process is usually described in the time domain. By using the sampling process, an analog signal is converted into a corresponding sequence of samples that are usually spaced uniformly in time. Clearly, for such a procedure to have practical utility, it is necessary that we choose the sampling rate properly, so that the sequence of samples uniquely defines the original analog signal. This is the essence of the sampling theorem. The sampling rate must be sufficiently large so that the analog signal can be reconstructed from the samples with sufficient accuracy. The sampling theorem, which is the basis for determining the proper sampling rate for a given signal, has a deep significance in signal processing

3. The Sampling Theorem

4. To combat the effects of aliasing in practice, we may use two corrective measures, as described here: 1. Prior to sampling, a low-pass anti-aliasing filter is used to attenuate those high frequency components of the signal that are not essential to the information being conveyed by the signal. 2. The filtered signal is sampled at a rate slightly higher than the Nyquist rate. some aliasing is produced by the sampling process if the sampling frequency is less than Nyquist rate. Aliasing refers to the phenomenon of a high-frequency component in the spectrum of the signal seemingly taking on the identity of a lower frequency in the spectrum of its sampled version, as illustrated in Figure 3.3.

5. The fact that the reconstruction filter has a well-defined transition band means that it is physically realizable. The reconstruction filter is a low-pass filter with A passband extending from - W to W, which is itself determined by the anti-aliasing filter. A transition band extending (for positive frequencies) from W to f s - W, where f s is the sampling rate.

6. Signal Reconstruction: The Interpolation Formula

8. Practical Sampling In proving the sampling theorem, we assumed ideal samples obtained by multiplying a signal g(t) by an impulse train which is physically nonexistent. In practice, we multiply a signal g(t) by a train of pulses of finite width, shown in Fig. b. The sampled signal is shown in Fig.c. Now is it possible to recover or reconstruct g(t) from the sampled signal g(t) in Fig. c?. Unsurprisingly, the answer is positive, provided that the sampling rate is not below the Nyquist rate. The signal g(t) can be recovered by low-pass filtering g(t) as if it were sampled by impulse train.