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NOTES-Chapter2

NOTES-Chapter2 - Some Useful Summations 1 Geometric...

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Some Useful Summations 1. Geometric Progression X i =0 a i = 1 1 - a | a | < 1 2. N X i =0 a i = 0 if a = 0 N + 1 if a = 1 1 - a N +1 1 - a otherwise 3. X i =0 ia i = a (1 - a ) 2 | a | < 1 (Take the derivative in Example 1 above) 4. N X i =1 i = N ( N + 1) 2 5. N X i =1 i 2 = N ( N + 1)(2 N + 1) 6 6. Sinc Function sinc x = sin x x 7. Taylor Series Expansion of an Exponential e x = X i =0 x i i ! 1

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ECE161A – Discrete Time Linear Systems 1. A second course in signal processing. 2. Have seen discrete time signals and systems as well as Z-transforms. 3. Will study 2 new transforms: DTFT, DFT 4. Will use Matlab for signal processing applications 5. Will discuss filters and will filter signals in Matlab 6. Will learn Fast Fourier Transform (FFT) which is used in many practi- cal applications such as spectrum analyzer, EKG systems, compression systems, etc. 7. In summary, we will cover an introduction to the important area of DIGITAL SIGNAL PROCESSING. Definition: A discrete time signal is one that is defined only for discrete points in time (hourly, every second, etc.) Ex.: An image on a computer is a discrete signal. It is defined only at discrete points in space, called pixels. Ex.: Any signal on a computer which is a list of numbers is a discrete signal. Ex.: A picture taken with a digital camera is a discrete signal. Ex.: A DVD format movie is a discrete signal. Ex.: An MP3 file is a discrete signal. 2
Chapter 2 – Discrete-Time Signals and Systems We assume that we derived a discrete-time signal from a continuous time signal via sampling . Given x a ( t ) to be a continuous time signal, x a ( nT ) is the value of x a ( t ) at t = nT . The discrete-time signal x [ n ] is defined only for n an integer. So if we derive x [ n ] from x a ( t ) by sampling every T seconds, where T is the sample period, we get: x a ( nT ) = x a ( t ) | t = nT x [ n ] = x a ( nT ) = x a ( t ) | t = nT We will not necessarily assume that x [ n ] is a discrete amplitude signal. A signal that is both discrete time and discrete amplitude is known as a digital signal. You will see these in later communications courses but a well-known example of a digital signal is music on a compact disk. Note that a discrete-time signal need not be generated by explicitly sam- pling a continuous-time signal. Some signals are inherently discrete time, such as computer bit sequences, and some signals are implicitly sampled, such as the daily DJIA or yearly temperature averages. Discrete-Time Signals and Systems Remember the square brackets! Read (skim) the example of Euler integration in the book but we will cover difference equations in Chapter 10 and when we discuss the Z-transform. Euler integration approximates the area under a curve x ( t ) by the sum of rectangular areas. 3

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Discrete-Time Unit Step Function u [ n ] = 1 , n 0 0 , n < 0 Notice that here, the unit step is defined at n = 0, unlike for continuous time.
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NOTES-Chapter2 - Some Useful Summations 1 Geometric...

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