Digital Signal Processing - Discrete-Time Systems Electrical Engineering and Computer Science University of Tennessee, Knoxville August 25, 2015
Overview 1 2 3 4 5 6 7 8
Introduction Three components of DSP Clarifications: Discrete-time vs. Digital Discrete-time signals unit sample (impulse), unit step, exponential sequences relationships periodicity of sinusoidal sequences
A general system and an LTI system y[n] = T {x[n]}
Linearity A system is linear iff T {ax x(n) 1 [n] + bx(n-n 2 [n]} 0 ) = at {x w(n) 1 [n]} + bt {x 2 [n]} Delay by n 0 System System Homogeneity or scaling or multiplicative property Additivity property x 1 (n) x 2 (n) α X β X x 1 (n) x 2 (n) + x(n) System System System y(n) y 1 (n) y 2 (n) y(n) α X β X Delay y(n-n 0 ) by n 0 + w(n) w[n] needs to be equal to y[n]
Time-invariance (or shift-invariance) For time-invariant systems, the input-output characteristics do not change over time, i.e., x[n] y[n] implies x[n n 0 ] y[n n 0 ] x(n) Delay by n 0 x(n-n 0 ) System System w(n) y(n) Delay y(n-n 0 ) by n 0 w[n] needs to be equal to y[n n 0 ] x 1 (n) x 2 (n) α x 1 (n) System System α y 1 (n) X β y 2 (n) X + w(n)
The convolution sum The unit sample uniquely characterizes the system. If δ[n] h[n], then δ[n k] h[n k]. x[n] = x[k]δ[n k] y[n] = x[k]h[n k] = x[n] h[n] Convolution sum is commutative. y[n] = r x[n r]h[r] = h[n] x[n]
Stability A system is said to be bounded-input bounded-output (BIBO) stable iff every bounded input produces a bounded output. i.e., there exist some finite numbers, B x and B y, s.t. for all n. x[n] B x <, y[n] B y < Abosolutely summable: For LTI, if x[n] <, to have y[n] <, we need to have k= h[k] <
Causality The output of a causal system at any time n depends only on present and past inputs, i.e., x[n], x[n 1], x[n 2],, but not on future inputs, i.e., x[n + 1], x[n + 2],
Static (or memoryless) vs. Dynamic (or with memory) systems For a static system, at any instant n, the output of the system only depends, at most, on the input sample at the same time: y[n] = T (x[n])
Exercises y[n] = x[2n] y[n] = x[ n] y[n] = ax[n] + b y[n] = e x[n] y[n] = x[n] cos ω 0 n, h[n] = 2 n u[ n] h[n] = 2 n u[n] h[n] = ( 1 2 )n u[n]
Linear constant-coefficient difference equation () Nth order N a k y[n k] = k=0 M b m x[n m] m=0 When N = 0, a 0 = 1, i.e., 0-th order M y[n] = b k x[n k] That is, h[n] = When N 0, a 0 = 1 k=0 { bn, n = 0, 1,, M 0, otherwise = M b k δ[n k] k=0 y[n] = M bm x[n m] N ak y[n k]
Initial conditions The homogeneous solution, y h [n] y[n] = y p [n] + y h [n] N a k y h [n k] = 0 y h [n] = k=0 N A m zm n m=1 If a system is characterized by an and is further specified to be linear, time-invariant, and causal, then the solution is unique. These auxiliary conditions are referred to as the initial-rest conditions. That is, if x[n] = 0 for n < n 0, then y[n] = 0 for n < n 0. Note that not for every set of boundary conditions that would correspond to an LTI system.
Exercises y[n] ay[n 1] = x[n] Suppose x[n] = δ[n] assume y[n] = 0, n < 0 assume y[n] = 0, n > 0
Eigenfunctions for LTI systems Complex exponentials are eigenvalues of LTI systems Let x[n] = e jωn, then y[n] = h[k]x[n k] = k= Let H(e jω ) = k h[k]e jωk, then y[n] = H(e jω )e jωn of the system: H(e jω ) = h[k]e jωk k k= h[k]e jω(n k) = e jωn k h[k of frequency : Function of continuous variable ω Periodic function of ω with period of 2π Generalization of frequency : the Fourier transform
Exercises Find the output to x[n] = A cos(ω 0 n + φ) Find the frequency to y[n] ay[n 1] = x[n] Find the output to x[n] = e jωn u[n] (suddenly applied exponential input)
Fourier-series expansion H(e jω ) is a continuous periodic function and would have a Fourier series representation H(e jω ) has a Fourier-series expansion in terms of complex exponentials The Fourier series coefficients then become the values of the unit sample H(e jω ) = n h[n]e jωn h[n] = 1 π H(e jω )e jωn dω 2π π
Fourier Transform -domain representation of arbitrary sequence x[n] X(e jω ) = n= x[n]e jωn x[n] = 1 π X(e jω )e jωn dω 2π π
The convolution property x[n] y[n] X(e jω )Y (e jω ) e jω0n H(e jω 0 )e jω 0n A k e jω k n A k H(e jω k )e jω k n k k x[n] = 1 π X(e jω )e jωn dω 2π π A decomposition of x[n] as a linear combination of complex exponentials y[n] = 1 2π π π X(e jω )H(e jω )e jωn dω Y (e jω ) = X(e jω )H(e jω ) y[n] = x[n] h[n]
The symmetry property For real sequences x[n], the symmetry property indicates that the of x[n] is a conjugate symmetric function of ω Other useful : X(e jω ) = X (e jω ) The real part of the is an even function: X R (e jω ) = X R (e jω ) The imaginary part of the is an odd function: X I (e jω ) = X I (e jω ) The amplitude of the is an even function: X(e jω ) The phase of the is an odd function X(e jω )