Course Outline (Tentative) Fundamental Concepts of Signals and Systems Signals Systems Linear Time-Invariant (LTI) Systems Convolution integral and sum Properties of LTI Systems … Fourier Series Response to complex exponentials Harmonically related complex exponentials … Fourier Integral Fourier Transform & Properties … Modulation (An application example) Discrete-Time Frequency Domain Methods DT Fourier Series DT Fourier Transform Sampling Theorem Laplace Transform Z Transform

Chapter II Chapter II Linear Time-Invariant (LTI) Systems

Linear Time-Invariant Systems Important class of systems as many physical phenomena can be modeled as such Representation of DT Signals in Terms of Pulses DT unit impulse can be used to construct any DT signal Think of a DT signal as a sequence of individual impulses Consider x[n]

Linear Time-Invariant Systems x[n] is actually a sequence of time-shifted and scaled impulses -2 x[-2]δ[n+2] x[-1]δ[n+1] 0 x[0]δ[n] 1 x[1]δ[n-1] 0 2 x[2]δ[n-2] 10

Linear Time-Invariant Systems It is possible to re-generate an arbitrary signal by sampling it with shifted unit impulse: This is called as sifting property: Sifting property of DT impulse weights shifted impulse x[n] = x[-4] δ[n+4] + x[-3] δ[n+3] + x[-2] δ[n+2] + + x[-1] δ[n+1] + x[0] δ[n] + x[1] δ[n-1]

DT Unit Impulse Response x[n] is superposition of scaled & shifted impulses The output of a linear system to x[n] = superposition of the scaled responses of the system to each of these shifted impulses Let h k [n] be the response of a linear system to δ[n-k]. From superposition property of linear systems, the response y[n] is the sum of individual responses, i.e.,

DT Unit Impulse Response In general, if the system is also time-invariant, then h k [n], response to δ[n-k], is a time-shifted version of h 0 [n], which is the response of the system to δ[n]; So, for k=0, i.e., input is δ[n], then the response of the system is Unit Impulse Response h[n] is the output (response) of the LTI system when δ[n] (impulse) is the input

Convolution Sum Representation of LTI Systems Hence, for an LTI system, the output y[n] for the input x[n] is Convolution Sum Superposition Sum convolution of the sequences (signals) x[n] and h[n], and represented by: Recall that for an LTI system (due to superposition property):

Convolution Sum Example Example: Consider an LTI system with impulse response h[n] and input x[n] is applied to the system. Find the output y[n]. 2 x[n] h[n] n h[n] + 21 y[n] h[n-1] 3

Convolution Sum Example Example: Consider LTI system with h[n] 0.5 x[n] h[n] n Consider h[n-k] as a time-reversed and shifted version of h[k] If we plot h[-k] then we can obtain h[n-k] simply by shifting to right (by n) if n>0, or to left if n<0 y[n]= x[k] h[n-k]

h[0-k] h[1-k] h[2-k] 120 k h[3-k] h[n-k], n>3 y[n]=0 0.5 x[k] 2 k 0 1 Convolution Sum Example Flip h[k] By shifting h[k] for all possible values of n, pass it through x[n]. h[n-k], n<0 0-2 k 01 k 1 n-1n n-2 k 0 k k n-1nn y[n]

Convolution Sum Example Ex: Consider x[n]= u[n] for 0< <1 and h[n]=u[n]. Find y[n]. h[-k] k y[n]= x[k] h[n-k] x[k]= u[k] k

Convolution Sum Example for n≥0 for n<0x[k]h[n-k]=0for k h[n-k] k n x[k]= u[k] k for n≥0x[k]h[n-k]≠0 for 0≤k≤n

Convolution Sum Example x[k]h[n-k] =

Convolution Sum Example Ex: Consider an LTI system with input x[n]=2 n u[-n] and h[n]=u[n]. Find y[n]. x[k] =2 k u[-k] k h[n-k] k n for n≥0, x[k]h[n-k] has nonzero samples for k≤0 y[n] = x[k]h[n-k] = 2 k = = = 2

Convolution Sum Example for n<0, x[k]h[n-k] has nonzero samples for k≤n y[n] y[n] = x[k]h[n-k] = 2 k =

Representation of CT Signals in terms of Impulses Consider a pulse or “staircase” approximation,, to a CT signal x(t) t x(t) -Δ-Δ 2Δ2Δ Δ kΔkΔ 0 Similar to DT case, can be expressed as a linear combination of delayed pulses Define 0 Δ δ Δ (t) t

Representation of CT Signals in Terms of Impulses Since has unit amplitude, we can represent in terms of shifted and scaled rectangular pulses… As Δ 0, approaches to x(t) (sifting property)

Representation of CT Signals in Terms of Impulses Other derivations for this ?? 1 Recall: Integrate both sides over 

CT Unit Impulse Response & Convolution Integral Let us define as the response of a linear system to As approaches to y(t) y(t) = lim (note as hence ) Hence,

CT Unit Impulse Response & Convolution Integral By time-invariance and therefore, impulse response! Convolution integral An LTI system is completely characterized by its impulse response (i.e., h[n] in DT, h(t) in CT)

Convolution Integral Example Let, a>0 be the input to an LTI system with h(t) = u(t). Find y(t)!

Convolution Integral Example 0 0 t for t<0 0 t for t>0 For t<0, y(t)=0 t>0

Convolution Integral Example for t>0

Properties of Convolution and LTI Systems 1) Commutative Property: Convolution is a commutative operation in both DT and CT. Easily shown by substituting variables Practical use ?? (take h[k], time-reverse and shift x[k] whenever it’s easier)

Properties of Convolution and LTI Systems 2) Distributive Property: Convolution distributes over addition. h 1 +h 2 xy h1h1 h2h2 + yx Practical interpretation ?? (parallel interconnection of systems)

Properties of Convolution and LTI Systems Practical ?? - (response to sum of two inputs = sum of responses to inputs individually) - Break a complicated convolution into several simpler ones (useful !!)

Properties of Convolution and LTI Systems 3) Associative Property: h 1 [n]h 2 [n] x[n]y[n] x[n] h 1 [n]*h 2 [n] y[n] x[n] h 2 [n]*h 1 [n] y[n] h 2 [n]h 1 [n] x[n]y[n] Result 1 by commutativity by associativity : (x[n]* h 1 [n])* h 2 [n] Result 2

Properties of Convolution and LTI Systems Results: Impulse response of cascade LTI systems is the convolution of their individual impulse responses Overall system response is independent of the order of the systems in cascade ! 4) LTI Systems with & w/o Memory: If then the LTI system is memoryless ! (look at the convolution sum ) Hence,memoryless

Properties of Convolution and LTI Systems 5) Invertibility of LTI Systems w(t)=x(t) h(t) h 1 (t) x(t) y(t) x(t) Identity System δ (t) x(t) if the system is invertible then and is the impulse response of the LTI inverse system.

Properties of Convolution and LTI Systems Example Consider LTI system of time-shift: y(t)=x(t-t 0 ) Impulse response of the system is h(t)= δ (t-t 0 ) Convolution of a signal with a shifted impulse shifts the signal Inverse: shift the output back!

Properties of Convolution and LTI Systems Example Consider an LTI system with h[n]=u[n] ; as u[n-k] = 0 for n-k<0 LTI inverse: y[n]=x[n]-x[n-1] (1 st difference) h 1 [n] = δ [n]- δ [n-1] Check if really inverse: h[n]*h 1 [n] = u[n]*( δ [n]- δ [n-1]) = u[n]* δ [n]-u[n]* δ [n-1] = u[n]-u[n-1] = δ [n] accumulator

Properties of Convolution and LTI Systems 6) Causality for LTI Systems: ; the output y[n] must not depend on x[k] for k>n h[n-k] should be zero for k>n h[n]=0 for n<0 h(t)=0 for t<0 Practical intuition ?? (impulse response must be zero before the impulse occurs: initial rest) For causal LTI systems, convolution sum becomes:

Properties of Convolution and LTI Systems In CT: h(t)=0 for t<0 causal; convolution integral becomes: REMARK: Causal Signal ! (zero for n<0, t<0)

Properties of Convolution and LTI Systems 7) Stability for LTI Systems: if the impulse response is absolutely summable then the system is stable! <B Consider

Properties of Convolution and LTI Systems In CT: if the impulse response is absolutely integrable then the system is stable! Example: Shift h(t)= δ (t-t 0 ) stable! Accumulator: h[n]=u[n] unstable!

Unit Step Response Another signal used quite often to describe the behaviour of LTI systems corresponds to the output when x[n]=u[n] or x(t)=u(t) : the step response of a DT LTI system is the running sum of its impulse response In continuous-time:

Unit Step Response : the step response of a CT LTI system is the running integral of its impulse response The impulse response is the first derivative of the unit step response

Causal LTI Systems Described by Differential and Difference Equations CT systems for which the input and output are related through a linear constant-coefficient differential equation, e.g., RC circuit DT systems for which the input and output are related through a linear constant-coefficient difference equation, e.g., bank account

Linear Constant-Coefficient Differential Equations Consider Implicit representation of input-output relationship Complete solution has two parts: - particular solution, y p (t) - homogeneous solution, y h (t) y h (t) is the solution to x(t)=0, i.e., y p (t) depends on x(t)

For x(t)=Ke 3t u(t)  y p (t)=Ye 3t for t>0 Substitute y p (t) and try   y p (t) =, t>0 y h (t)=Ae st  Ase st +2Ae st =0  s=-2  y h (t)=Ae -2t  y(t)=, t>0 Linear Constant-Coefficient Differential Equations

 We need initial conditions.  For causal LTI systems, we assume initial rest, y(0)=0  substitute:  Linear Constant-Coefficient Differential Equations

In general, N th order linear constant-coefficient differential equation is  has particular solution + homogeneous solution  initial (auxiliary) conditions are necessary.  for causal LTI systems, initial rest:

Linear Constant-Coefficient Differential Equations Assume an LTI system described by N th order linear constant- coefficient differential equation as follows  In this case, we find that h(t) (impulse response) satisfies  with the initial (auxiliary) conditions (recall that for causal LTI systems, initial rest): Check Auxiliary text 2 (Yuksel) for proof.

Linear Constant-Coefficient Differential Equations Example: Consider the LTI system described by Let us find the impulse response first. h(t) satisfies The characteristic equation is:  Satisfied for s=-3, and s=-4 When the initial conditions are used, the coefficients are determined as You can then obtain the output by using convolution

Linear Constant-Coefficient Difference Equations In DT: Rearrange it, then: Recursive operation as the output at time n is a function of previous values of input and output. We need initial conditions: y[-1],..., y[-N]

Linear Constant-Coefficient Difference Equations As a special case, N=0, then  Its impulse response:  This is a finite duration impulse response. Such systems are called finite impulse response (FIR) systems. (non-recursive (LTI system), No need for initial conditions)

Linear Constant-Coefficient Difference Equations Example: Consider the difference equation  recursive; initial conditions are needed Assume and initial rest, i.e., x[n]=0, y[n]=0 for n≤-1:  for K=1, For N≥1: Infinite Impulse Response (IIR)

Block Diagram Representations y[n]+ay[n-1]=bx[n] (addition, multiplication, delay) + D x 1 [n] x 2 [n] x 1 [n]+ x 2 [n] x[n] a ax[n]x[n]x[n-1] y[n]=-ay[n-1]+bx[n]: + D x[n] b -a y[n-1] y[n] Used to help in understanding and implementation of systems DT Case: First-order difference equation

Block Diagram Representations CT Case: First-order differential equation + D x(t) b/a -1/a y(t)

Block Diagram Representations In real implementations, differentiators are difficult to implement and extremely sensitive to error & noise. + ∫ x(t) b -a y(t) (For initial rest, take integral from -∞ to t) OP-AMP ! ∫ x(t)