Information Transmission Chapter 2, What is bandwidth OVE EDFORS - PowerPoint PPT Presentation

Information Transmission Chapter 2, What is bandwidth OVE EDFORS ELECTRICAL AND INFORMATION TECHNOLOGY

L e a r n i n g o u t c o m e s ● A f t h e r t h i s l e c t u r e , t h e s t u d e n t s h o u l d – u n d e r s t a n d t h e b a s i c p r i n c i p l e s o f s a m p l i n g , i n c l u d i n g – t h e c o n c e p t o f o r t h o g o n a l b a s i s f u n c t i o n s , – t h e s a m p l i n g t h e o r e m , – N y q u i s t r a t e s / f r e q u e n c i e s a n d S h a n n o n b a n d w i d t h s , a n d – b e a b l e t o p e r f o r m c a l c u l a t i o n s o n n e c e s s a r y s a m p l i n g r a t e s b a s e d o n t h e c h a r a c t e r i s t i c s o f t h e s a m p l e d s i g n a l s . O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 2

Wh e r e a r e w e i n t h e B I G P I C T U R E ? Electronics for analog input Models of transmission and Lecture relates to pages and output, including storage media. 58–66 in textbook. sampling and reconstruction. O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 3

“Certain factors affecting telegraph speed”, H. Nyquist, The Bell System Technical Journal (Volume:3 , Issue: 2 ), 1924 “This paper considers two fundamental factors entering into the maximum speed of transmission of intelligence by telegraph. These factors are signal shaping and choice of codes. The first is concerned with the best wave shape to be impressed on the transmitting medium so as to permit of greater speed without undue interference either in the circuit under consideration or in those adjacent….” O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 4

A n i n t r o d u c t o r y e x a m p l e Assume Combining with the time scaling property, yields What is the bandwidth of the signal? O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 5

Wh a t i s t h e b a n d w i d t h o f a r e c t p u l s e ? The spectrum is not confined to a finite band O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 6

B a n d w i d t h o f a s i g n a l A softer restriction of bandwidth, that works for all signals, is that the signal x(t) has a certain fraction of its energy inside the frequency band [- W , W ], i.e. With this it seems reasonable to say that the signal has a Fourier bandwidth of W = 1/T Hz. To be more precise, we could say that the “90%-energy Fourier bandwidth'' of the signal is W = 1/T Hz, but we will usually not have need for such precision. O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 7

WL A N - I E E E 8 0 2 . 1 1 s p e c t r u m m a s k 802.11b 802.11a Source: http://www.rfcafe.com/references/electrical/wlan-masks.htm O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 8

The sampling theorem O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 9

“Certain Topics in Telegraph Transmission Theory”, H. Nyquist, Transactions of the American Institute of Electrical Engineers (Volume:47 , Issue: 2 ), 1928 “The most obvious method for determining the distortion of telegraph signals is to calculate the transients of the telegraph system... The present paper attacks the same problem from the alternative standpoint…This method of treatment necessitates expressing the criteria of distortion- less transmission in terms of the steady-state characteristics. Accordingly, a considerable portion of the paper describes and illustrates a method for making this translation. A discussion is given of the minimum frequency range required for transmission at a given speed of signaling….” O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 1 0

B a s i s f u n c t i o n s f o r s a m p l i n g Consider the function with Fourier transform is confined to the frequency band [ -W,W ]. O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 1 1

O r t h o n o r m a l f u n c t i o n s Versions of delayed by , where k is an integer, form a set of orthonormal functions . The term orthonormal means that the functions are orthogonal and normalized . O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 1 2

O r t h o g o n a l f u n c t i o n s Orthogonality is an important notion in signal analysis. it means that where is the energy of , i.e., O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 1 3

O r t h o g o n a l i t y o f t h e b a s i s f u n c t i o n s , where k is an integer, are orthogonal functions since Parceval's theorem What does this mean? O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 1 4

N o r m a l i z e d f u n c t i o n s Furthermore, since for all k , these functions are normalized (energy ). A set of orthogonal and normalized functions is called an orthonormal set of functions . O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 1 5

Tie s a m p l i n g t h e o r e m If x(t) is a signal whose Fourier transform is identically zero for , then x(t) is completely determined by its samples taken every seconds in the manner See textbook for a proof O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 1 6

I l l u s t r a t i o n o f s a m p l i n g / r e c o n s t r u c t i o n O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 1 7

N y q u i s t r a t e / N y q u i s t f r e q u e n c y The sample points are taken at the rate (frequency) 2W samples per second. If W is the smallest frequency such that the Fourier transform of x(t) is identically zero for then the sampling rate 2W is called the Nyquist rate or Nyquist frequency . O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 1 8

S h a n n o n b a n d w i d t h Let denote the smallest such that is orthogonal to every time-shift of itself, , by a nonzero integer multiple of . We call the Nyquist-shift of the basis signal The Nyquist-shift of the signal is The Shannon bandwidth B of the basis signal is or equivalently basis functions per second O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 1 9

Tie f u n d a m e n t a l t h e o r e m o f b a n d w i d t h The Shannon bandwidth B of a basis signal is at most equal to its Fourier bandwidth W ; equality holds when the signal is a sinc function. The Shannon bandwidth can be thought of as the amount of bandwidth a signal needs and the Fourier bandwidth as the amount of bandwidth a signal uses . O v e E d f o r s E I T A 3 0 - C h a p t e r 2 ( P a r t 3 ) 2 0