Sampling theorem

ECE

1. SIGNALS AND SYSTEMS Presented by., S.Shanmathee Sampling Theorem

2. 2/6/2015 Why We need Sampling??

3. ADC Generally signals are analog in nature (eg:speech,weather signals). To process the analog signal by digital means, it is essential to convert them to discrete-time signal , and then convert them to a sequence of numbers. The process of converting an analog to digital signal is ‘Analog-to-Digital Conversion’. The ADC involves three steps which are: 1)Sampling 2)Quantization 3)coding 2/6/2015

4. Analog signals: continuous in time and amplitude Example: voltage, current, temperature,… Digital signals: discrete both in time and amplitude Example: attendance of this class, digitizes analog signals,… Discrete-time signal: discrete in time, continuous in amplitude Example: hourly change of temperature in Austin TYPES OF SIGNALS 2/6/2015

5. SAMPLING During sampling process, a continuous-time signal is converted into discrete -time signals by taking samples of continuous-time signal at discrete time intervals. T=Sampling Interval x (t)=Analog input signal 2/6/2015

6. SAMPLING THEOREM Sampling theorem gives the criteria for minimum number of samples that should be taken. Sampling criteria:- ” Sampling frequency must be twice of the highest frequency ” f s =2W f s =sampling frequency w=higher frequency content 2w also known as Nyquist rate 2/6/2015

7. Contd … Nyquist rate is defined as the minimum sampling rate for the perfect reconstruction of the continuous time signals from samples. Nyquist rate=2*highest frequency component =2*W So sampling rate must be greater than or equal to nyquist rate 2/6/2015

8. PROOF OF SAMPLING THEOREM There are two parts, representation of x(t) in its samples reconstruction of x(t) Representation of x(t) in its samples 1.Define x ∂ (t) 2.Take fourier transform of x ∂ (t)) ( i.e ) x ∂ (f) 3.Relation between x(f) and x ∂ (f) 4.Relation between x(t) and x( nT s ) 2/6/2015

9. Contd … Reconstruction of x(t) 1.Take inverse fourier transform of x ∂ (f) 2.Show that x(t) is obtained back with the help of interpolation function 2/6/2015

10. ALIASING While providing sampling theorem we considered f s =2W Consider the case that f s < 2W 2/6/2015

11. Contd … Effects of Aliasing, 1.Distortion. 2.The data is lost and it cannot be recovered. To avoid Aliasing, 1.sampling rate must be f s >=2W. 2.strictly bandlimit the signal to ’W’. 2/6/2015

12. PROBLEMS 2/6/2015

13. Determine Nyquist rate for continuous time signal In general form, any continuous signal can be written as S(t)=A 1 cos (jw 1 t)+ A 2 cos (jw 2 t)+ A 3 cos (jw 3 t) F 1 = w 1 /2∏ = 50∏/2∏ = 25HZ F 2 = w 2 /2∏ = 300∏/2∏ = 150HZ F 3 = w 3 /2∏ = 100∏/2∏ = 50HZ Here, highest frequency component=150HZ Hence Nyquist rate=2*150HZ=300HZ ANS: 2/6/2015

14. Given continuous time signal What is the minimum sampling rate(nyquist rate)? Highest frequency=100HZ So, Nyquist rate=2W=2*100=200HZ If sampling frequency is 400HZ then what is the discrete time signal obtained? f=freq of continuous signal/sampling freq =100/400=1/4 Discrete time signal=5 cos (2∏fn)=5 cos (2∏*1/4 n) =5 cos (∏n/2) 2/6/2015

15. 2/6/2015 REFERENCES “SIGNALS AND SYSTEMS” by Dr.J.S.Chitode

16. THANK YOU Optimist: "The glass is half full." Pessimist: "The glass is half empty." Engineer : "That glass is twice as large as it needs to be." 2/6/2015