Nyquist Sampling Theorem By: Arnold Evia Table of Contents • What is the Nyquist Sampling Theorem? • Bandwidth • Sampling • Impulse Response Train • Fourier Transform of Impulse Response Train • Sampling in the Fourier Domain o Sampling cases • Review What is the Nyquist Sampling Theorem? • Formal Definition: o If the frequency spectra of a function x(t) contains no frequencies higher than B hertz, x(t) is completely determined by giving its ordinates at a series of points spaced 1/(2B) seconds apart. • In other words, to be able to accurately reconstruct a signal, samples must be recorded every 1/(2B) seconds, where B is the bandwidth of the signal. Bandwidth
• There are many definitions to bandwidth depending on the application • For signal processing, it is referred to as the range of frequencies above 0 (|F(w)| of f(t)) Bandlimited signal with bandwidth B • Signals that have a definite value for the highest frequency are bandlimited (|F(w)|=0 for |w|>B) • In reality, signals are never bandlimited o In order to be bandlimited, the signal must have infinite duration in time Non-bandlimited signal (representative of real signals) Sampling
• Sampling is recording values of a function at certain times • Allows for transformation of a continuous time function to a discrete time function • This is obtained by multiplication of f(t) by a unit impulse train Impulse Response Train • Consider an impulse train: • Sometimes referred to as comb function • Periodic with a value of 1 for every nT0, where n is integer values from -∞ to ∞, and 0 elsewhere Fourier Transform of Impulse Train InputSet upthe Equations function into the SolveSolve Dn for for one Dn period SubstituteUnderstand Dn into Answerfirst equation fourier transform eqs. Consider period from – The fourier spectra of the function T0 is the period of the T0/2 to T0/2 has an amplitude of 1/T0 at nw0 func. Only one value: at t=0 for values of n from –∞ to +∞, and Integral equates to 1 as 0 elsewhere e-jnw0(0) = 1 Distance between each w0 is dependent on T0. Decreasing T0, increases the w0 and distance
Original Function Fourier Spectra Sampling in the Fourier Domain • Consider a bandlimited signal f(t) multiplied with an impulse response train (sampled): o If the period of the impulse train is insufficient (T0 > 1/(2B)), aliasing occurs o When T0=1/(2B), T0 is considered the nyquist rate. 1/T0 is the nyquist frequency Visual Representation of Property Time • Recall that multiplication in the time Domain domain is convolution in the frequency domain: . = • As can be seen in the fourier spectra, it is Freq. Domain only necessary to extract the fourier spectra from one period to reconstruct the signal! * = Sampling Cases • T0>1/(2B) o Undersampling o Distance between copies of F(w) that overlap happens o Aliasing occurs, and the higher frequencies of the signal are corrupted • T0<=1/(2B) o Oversampling o Distance between copies of F(w) is sufficient enough to prevent overlap o Spectra can be filtered to accurately reconstruct signal Review • Nyquist sampling rate is the rate which samples of the signal must be recorded in order to accurately reconstruct the sampled signal o Must satisfy T0 <= 1/(2B); where T0 is the time between recorded samples and B is the bandwidth of the signal • A signal sampled every T0 seconds can be represented as:
where Ts = T0 Review (cont.) • One way of understanding the importance of the Nyquist sampling rate is observing the fourier spectra of a sampled signal • A sampled signal’s fourier spectra is a periodic function of the original unsampled signal’s fourier spectra o Therefore, it is only necessary to extract the data from one period to accurately reconstruct the signal • Aliasing can occur if the sampling rate is less than the Nyquist sampling rate o There is overlap in the fourier spectra, and the signal cannot be accurately reconstructed (Undersampling) References
Some basic resources can be found here: • http://www.cs.cf.ac.uk/Dave/Multimedia/node149.html • http://www.youtube.com/watch?v=7H4sJdyDztI
ARC website: • http://iit.edu/arc/
ARC BME schedule: • http://iit.edu/arc/tutoring_schedule/biomedical_engineering.s html