EECS 20 N March 21, 2001 Lecture 26: Frequency Response (III) Laurent El Ghaoui 1
outline reading assignment: Chapter 8 of Lee and Varaiya we ll concentrate on continuous-time systems: convolution integral impulse impulse response 2
convolution sum in the discrete-time domain, the convolution sum of two signals x and h is defined by (x h)(n) = m=+ m= x(m)h(n m) what is the analog in the continuous-time domain? 3
convolution in continuous-time in continuous-time, we define the convolution integral of two signals x and h by (x h)(t) = + x(τ)h(t τ) dτ example: if h(t) = 1/3 if t [0 3] 0 otherwise then (h x)(t) = 1 3 3 0 x(τ) dτ i.e., the length-three continuous-time moving average! 4
commumtativity of convolution just like the convolution sum, the convolution integral is commutative, that is, x h = h x for every pair of signals x, h proof: + x(τ)h(t τ) dτ = + x(t τ )h(τ ) dτ by change of variables τ = t τ 5
convolution and systems given a signal h, we can define a system S by S(x) = (h x) this system is linear, and time-invariant: for any s, (Ds(h x))(t) = (h x)(t s) = + x(t s τ)h(τ)dτ = (h Ds(x)) where Ds(x)(t) = x(t s) is the delay by s these properties are analogous to those of convolution in discrete-time 6
discrete-time impulse in the discrete-time domain, the unit impulse at n = 0 is defined by 1 if n = 0 δ(n) = 0 otherwise what is the analog in the continuous-time domain? 7
fundamental property of discrete-time impulse the fundamental property of the discrete-time unit impulse at n = 0 is that for any signal x, (δ x)(n) = m=+ m= = x(n) x(m)δ(n m) that is, x [Ints Reals], (δ x) = x 8
interpretations the impulse is the identity with respect to convolution (plays the same role as the identity matrix does with respect to matrix product) the above property uniquely defines the unit impulse: if a signal v satisfies x [Ints Reals], (v x) = x then v is the unit impulse: v = δ if a system S defined by a convolution sum with a given signal h, S(x) = h x then S is LTI, and h is the impulse response, since S(δ) = h δ = h 9
continuous-time impulse we ll define the unit impulse in the continuous-time domain a similar way, as a signal δ such that x [Reals Reals], (δ x) = x 10
interpretations the impulse is the identity with respect to convolution (plays the same role as the identity matrix does with respect to matrix product) the above property uniquely defines the unit impulse: if a signal v satisfies x [Reals Reals], (v x) = x then v is the unit impulse: v = δ if a system S defined by a convolution integral with a given signal h, S(x) = h x then S is LTI, and h is the impulse response, since S(δ) = h δ = h 11
what is a continuous-time impulse, anyway? it turns out that in the continuous-time domain, the unit impulse δ is not a function... despite its name, Dirac delta function we can only understand it as a limit 12
let approximating an impulse δɛ(t) = 1/2ɛ if t [ ɛ ɛ] 0 otherwise δɛ(t) PSfrag replacements 1 2ɛ t 2ɛ 13
interpretation of δɛ δɛ is a signal with very large values over a very brief time interval, such that the energy is one: + δɛ(t) dt = 1 14
impulse as a limit now let s convolve δɛ with a continuous-time signal x: (x δɛ)(t) = = 1 2ɛ + t+ɛ t ɛ x(τ)δɛ(t τ) dτ x(τ) dτ = average of x(τ) over [t ɛ t + ɛ] x(t) when ɛ 0 so, the limit δ = limɛ 0 δɛ can be understood as the unit impulse at t = 0 in the continuous-time domain 15
it s not a function we cannot say that δ is a function, because we cannot assign a finite value at t = 0!! indeed, when ɛ 0 the function 1/2ɛ if t [ ɛ ɛ] δɛ(t) = 0 otherwise takes on an infinite value at t = 0 we can say, however, that δ vanishes outside t = 0 (just like in discrete-time) 16
understanding the impulse the best way to understand the impulse is to think of it as a system, that is, an operator on signals precisely, as a convolution operator S : x S(x) = (h x), where h is such that S(x) = x for every signal x δ(t) PSfrag replacements 1 t 17
signals as sums of weighted delta functions given any signal x, we have by definition x = δ x that is, x(t) = we can interpret this as: + x(τ)δ(t τ) dτ any signal is the weighted sum of delayed impulses again, we had a similar result in the discrete-time domain (with finite sums) 18
impulse response for a continuous-time, linear, time-invariant system S, we define the impulse response by h = S(δ), that is, the (zero-state) response to a unit impulse at t = 0 physically, this corresponds to the response to a very brief signal with very large amplitude at t = 0 19
impulse response and convolution suppose we are given an LTI system, S what is the (zero-state) response to an arbitrary input x? fix τ Reals if the input x is the delayed impulse Dτ (δ), then by definition of the impulse response h = S(δ) time-invariance of S we obtain S(Dτ (δ)) = Dτ (S(δ)) = Dτ(h) 20
case of arbitrary input let x be now an arbitrary input from the decomposition x(t) = (δ x)(t) = + x(τ)δ(t τ) dτ and by linearity of S, we obtain the zero-state response: (S(x))(t) = + x(τ)h(t τ) dτ in short: S(x) = h x 21
summary the output of a continuous-time LTI system is given by the convolution of the input signal with the impulse response: S(x) = h x the impulse response is simply the response to a particular input called unit impulse, also called the Dirac delta function the Dirac delta function is not a function: it can be understood as the limit of functions that have very large values on a very small time interval, and have finite energy hence, the impulse response describes how an LTI system reacts to such idealized signals 22