# ECE 4371, Fall, 2014 Introduction to Telecommunication Engineering/Telecommunication Laboratory Zhu Han Department of Electrical and Computer Engineering.

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ECE 4371, Fall, 2014 Introduction to Telecommunication Engineering/Telecommunication Laboratory Zhu Han Department of Electrical and Computer Engineering Class 7 Sep. 17 th, 2014

Outline Analog vs. Digital ADC/DAC: gateway between analog and digital domains –Sampling Theorem –Quantization –Most important part in communication system –Most important during interview –Read books carefully Examples

Claude Elwood Shannon, Harry Nyquist

Sampling Theory In many applications it is useful to represent a signal in terms of sample values taken at appropriately spaced intervals. The signal can be reconstructed from the sampled waveform by passing it through an ideal low pass filter. In order to ensure a faithful reconstruction, the original signal must be sampled at an appropriate rate as described in the sampling theorem. –A real-valued band-limited signal having no spectral components above a frequency of B Hz is determined uniquely by its values at uniform intervals spaced no greater than seconds apart.

Sampling Block Diagram Consider a band-limited signal f(t) having no spectral component above B Hz. Let each rectangular sampling pulse have unit amplitudes, seconds in width and occurring at interval of T seconds. A/D conversion f(t) T f s (t) Sampling

Impulse Sampling

Impulse Sampling with increasing sampling time T

Introduction Equation number is not the same as in the book

Math

Math, cont.

Interpolation Formula

Interpolation If the sampling is at exactly the Nyquist rate, then

Practical Interpolation Sinc-function interpolation is theoretically perfect but it can never be done in practice because it requires samples from the signal for all time. Therefore real interpolation must make some compromises. Probably the simplest realizable interpolation technique is what a DAC does.

Sampling Theorem

Under Sampling, Aliasing

Avoid Aliasing Band-limiting signals (by filtering) before sampling. Sampling at a rate that is greater than the Nyquist rate. A/D conversion f(t)f(t) T f s (t) Sampling Anti-aliasing filter

Anti-Aliasing

Aliasing 2D example

Example: Aliasing of Sinusoidal Signals Frequency of signals = 500 Hz, Sampling frequency = 2000Hz

Example: Aliasing of Sinusoidal Signals Frequency of signals = 1100 Hz, Sampling frequency = 2000Hz

Example: Aliasing of Sinusoidal Signals Frequency of signals = 1500 Hz, Sampling frequency = 2000Hz

Example: Aliasing of Sinusoidal Signals Frequency of signals = 1800 Hz, Sampling frequency = 2000Hz

Example: Aliasing of Sinusoidal Signals Frequency of signals = 2200 Hz, Sampling frequency = 2000Hz

Natural sampling (Sampling with rectangular waveform) Figure 6.10

Bandpass Sampling (a) variable sample rate (b) maximum sample rate without aliasing (c) minimum sampling rate without aliasing

Bandpass Sampling A signal of bandwidth B, occupying the frequency range between f L and f L + B, can be uniquely reconstructed from the samples if sampled at a rate f S : f S >= 2 * (f2-f1)(1+M/N) where M=f 2 /(f 2 -f 1 ))-N and N = floor(f 2 /(f 2 -f 1 )), B= f 2 -f 1, f2=NB+MB.

Bandpass Sampling Theorem

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