Sampling process, Aliasing effect, Quantization

Sampling process, Aliasing effect, Quantization

1. ANALOG TO DIGITAL CONVERSION

2. Digital Signal Processing System Anti Aliasing Filter Sample + Hold A/D Converter DSP D/A Converter Reconstruction Filter Analog input signal Analog Output signal

3. Analog to Digital Converter Continuous- time Continuous- amplitude input signal SAMPLER QUANTIZER ENCODER Discrete-time Continuous- amplitude signal Discrete-time Discrete amplitude signal Digital Output signal

4. Sampling Process Sampling is the process by which continuous-time signal is converted into discrete-time signal

5. Let the sampled signal is represented by where g(t) is the sampling function using Fourier series, it is expressed as where where f s is the fundamental / Sampling frequency

6. Using Fourier transform the spectrum of x s (t) is denoted by Substituting x s (t) in the above equation, Thus by comparing with eqn 1

7. Spectrum of sampled signal The signal x(t) is assumed to have no frequency components above f h that is in the frequency domain .Such signal is said to be band limited. The frequency is equal to twice the highest frequency in x(t) that is 2f h is called the Nyquist rate Sampling Theorem: A band limited continuous time signal, with higher frequency f h Hz can uniquely recovered from its samples provided the sampling rate f s >2f h

8. Aliasing Effect Aliasing Effect: If we sample the x(t) with a sampling frequency f s <2f h the reconstruction of original continuous signal from its discrete-time signal by filtering is very difficult because of spectral overlap. The original shape of the signal is lost due to under sampling. This overlap is known as Aliasing.

9. Quantization The process of converting discrete time continuous signal into discrete time discrete amplitude signal Quantization step size Sampled Value x(n) (Decimal Reprs) Binary Representation Rounding Quantized Value x q (n) Quantization Error e(n) = x q (n)- x(n) 0.620 0.10011110 0.101 0.625 0.005

10. Illustration of Quantization

11. Sampling Technique Ideal or Instantaneous Sampling

12. Sampling Technique Natural or Chopper Sampling

13. Sampling Technique Flat top or Rectangular pulse Sampling

14. Elementary Signals Unit Step Unit Impulse Unit Ramp Exponential Complex exponential and sinusoidal signal

15. Unit Step

16. Unit Impulse

17. Unit Ramp

18. Exponential

19. Complex exponential and sinusoidal Exponential Sinusoidal