Signal Processing and Linear Systems Lecture : Characterizing Systems Nicholas Dwork www.stanford.edu/~ndwork Our goal will be to develop a way to learn how the system behaves. In general, this is a very difficult thing to do. We will see that if our system has some of the nice properties we ve already discussed, then this becomes remarkably easy.
Outline Discrete Systems Continuous Systems Discrete Systems Consider a system that accepts a vector and outputs a vector. f S g We want to learn how this system behaves. How can we do that? In general, we have to check every vector that we re interested in. That can take a long (perhaps infinite) amount of time.
Any vector can be written as Suppose we know that our system is linear. That helps a lot! Consider the example where S : C! C v C v + v + v What is the output when our system is applied to this vector? S 8 < : v + v + v 9 ; S 8 < : v 9 ; + S 8 < : v 9 ; + S 8 < : v 9 ; v S 8 < : 9 ; + v S 8 < : 9 ; + v S 8 < : 9 ; Now we don t need to know the output of every vector. We only need to know the output from three vectors!!! 6
In general, for a linear system S : C M! C N S{v} MX v i S{e i } i This is called the superposition summation. This shows that we only need to know the outputs for order to know the output for any vector. M vectors in 7 What if the system S is also shift invariant? These are called Linear Shift Invariant (LSI) systems. 8 9 8 >< > >< S S 6 7. >: >; >: 8 99 >< > > 6 7. >: >; >; 8 8 99 >< >< > > S 6 7. >: >: >; >; In general S{e i } S{ i {e }} i {S{e }} 8
S{e i } S{ i {e }} i {S{e }} Recall the superposition summation S{v} MX v i S{e i } i Let s combine these two: S{v} MX v i i {S{e }} i We only need to know the output for vector to determine the output of any vector!!! 9 S{v} MX v i i {S{e }} i e is so important that we give it a special name and symbol. e is called the Kronecker impulse (or delta) function. S{e } is so important that we give it a special name and symbol. h S{e } is called the impulse response. NX S{v} v i i {h} With these symbols, i
Suppose Circular Convolution f,g C N. Then the circular convolution of f and g is f ~ g NX f i i {g} i NX (f ~ g)[n] f[m] g [(n m) mod N ] m This shows that for an LSI system NX S{v} v i i {h} v ~ h. i Major Theorem: For an discrete LSI system, the output is equal to the input circularly convolved with the impulse response.
Outline Discrete Systems Continuous Systems Continuous Systems Consider a system that accepts a function and outputs a function. f S g We want to learn how this system behaves. How can we do that? In general, we have to check every function that we re interested in. That can take a long (perhaps infinite) amount of time.
Suppose we know that our system is linear. That helps a lot! But first, we need a new tool. With discrete systems we had a vector x NX i i x i x such that With continuous systems, we need an analogous function Z (s) x(s)ds x() Impulse Function The Dirac Delta Function Z is the function that satisfies the following: (s) x(s)ds x() Basic idea: acts over a time interval which is very small. During this small time interval, x(s) x(). 6
Impulse Function Intuition Approximate as g n (s) n (ns). (s) The area of g n is n (/n). g n (s) x(s) g n (s) g (s) g (s) x() - - Z Z Z x(s) (s) ds lim x(s) g n (s) ds x() g n (s) ds x() n! 7 Scaled Impulses is an impulse at time with magnitude or strength or area. Z (s) x(s)ds x() On plots, draw an arrow and write the strength next to the arrow. Ex: 8
Scaling the Argument of Impulses (as) a (s) Again, this can be proved with variable substitution. 9 Sifting Property When the impulse function is shifted, we get a shifted value of. Z x(s) (s ) ds x( ) This can be proved by a variable substitution. x x(s) g n (s ) x( )
Now that we have the impulse function, how can we characterize a continuous linear system? Before, we noted that any vector v v C could be written as + v + v We need to write a function x similarly. It turns out, now we can! x( ) Z x(s) ( s) ds where h Z S {x( )} S Z Z Z x(s) ( s) ds S {x(s) ( x(s) S { ( x(s) h (s) ds s)} ds s)} ds is the response of the system applied to an impulse at location. This is called the Superposition integral.
What if the system is also shift invariant? Z S{x( )} h Z x(s) S { ( x(s) h( s)} ds s) ds where is the response of the system to an impulse function (the impulse response). To determine the output of an LSI system to any function, we only need to know how the system responds to one function - an impulse!!! The convolution of f and g is (f g)( ) Convolution Z f(s) g( s) ds With this notation, S{x} x h Major theorem: For a continuous LSI system, the output is equal to the input convolved with the impulse response.
(f g)( ) Z Graphical Interpretation f(s) g( s) ds f f Let s consider a specific. s g g( s) s (f g)( ) Z Graphical Interpretation f(s) g( s) ds f f Let s consider a specific. s g( s) g( s) s s 6
(f g)( ) f Z Graphical Interpretation f(s) g( s) ds Let s consider a specific. f(s) g( s) g( s) s s x And then we integrate to get the area under the curve. That gives us the value of (f g)( ) s 7 Flip and Slide 8
9 Examples of System Responses
Twisted Pair Cable The output is delayed and smoothed. Transmitting. bits / sec. We can easily see the s and s in the output. Transmitting bits / sec. The signal appears lost.
Multi-path Echoes High Energy Photon Detectors Can be modeled as having a simple exponential decay impulse response. They are used in Positron Emission Tomography (PET) and Nuclear Scintigraphy medical imaging modalities.
Input: a series of photons (modeled as impulses) Output: recorded light is a superposition of impulse responses. Try these: 6
Properties of Convolution Associative Commutative Distributive f (g h) (f g) h f g g f f (g + h) (f g)+(f h) Identity f f 7 Multiple Systems What if we had multiple systems? x S S S n y Even if we knew the impulse responses, to estimate the output for a given input, we would have to perform n convolutions!!! That would be hard and boring. There s a better way 8