ECE3123_Chapter-02_Discrete_Time_Systems - ECE3123 Digital Signal Processing CHAPTER 2 DISCRETE-TIME SIGNALS SYSTEMS Prof Dr Othman O Khalifa Electrical

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Unformatted text preview: ECE3123 Digital Signal Processing CHAPTER 2 DISCRETE-TIME SIGNALS & SYSTEMS Prof. Dr. Othman O. Khalifa Electrical and Computer Engineering Kulliyyah of Engineering International Islamic University Malaysia Continuous-Time Sinusoidal Signals A simple harmonic oscillation is mathematically described by the following continuous-time sinusoidal signal: The subcript a used with x(t) denotes an analog signal which is characterized by; the sinusoid amplitude (A), frequency (F) in cycles per second or hertz (Hz), and phase (θ) in radians. 8/5/2009 Prof. Dr Othman O Khalifa ٢ ١ Discrete-Time Sinusoidal Signals A discrete-time sinusoidal signal may be expressed as x(n) = A cos(2πfn + θ ), −∞ < n < ∞ Use ω = 2πf to alternatively convert to radians per sample where n is the sample number, A is the sinusoid amplitude, f is the frequency in cycles per sample, and θ is the phase in radians. 8/5/2009 ٣ Classification of signals Multichannel (signals are generated by multiple sensors/sources which are measuring the same parameter. Example: measurement of ground acceleration a few kilometers from epicenter of an earthquake, ECG etc) Multidimentional (The value of signal is a function of M independent variables.) 8/5/2009 Prof. Dr Othman O Khalifa ٤ ٢ Continuous time Vs Discrete time (may not be discrete valued) Continuous valued Vs discrete valued (quantized continuous valued discrete time signal) Deterministic Vs Random (Deterministic, if signal can be uniquely described by an explicit mathematical expression e. g. sine, cosine, line, parabola etc. Random signal examples noise, unsynchronized digital signal etc ) 8/5/2009 ٥ For Discrete time sinusoids x(n) = Acos(ωn+φ); ω = 2πf; f (cycles per sample) = F/Fs; Fs is the sampling freq Periodic, if ‘f’ is a rational number. for periodicity, x(n+N) = x(n) for all n i.e. Acos(2πf(n+N)+φ) = Acos(2πfn+φ) – This relation is true if and only if there exists an integer K such that 2πfN = 2πk i.e. f=k/N – If k and N are relatively prime then N is called the fundamental period of x[n]. – A small change in freq can results in large change in period i.e. f1= 31/60 implies N1=60 but f2=30/60 implies N2= 2. 8/5/2009 Prof. Dr Othman O Khalifa ٦ ٣ Signal Types and Properties Energy Vs Power signals – Energy of x(n) = E= ∑-∞<n<∞|x(n)|2 <n<∞ – If E is finite, x(n) is called energy signal. – Power of x(n) = P = lim(N tend to ∞) {1/(2N+1)}* {∑- N<n<N|x(n)|2} If E is finite P=0 but if E is infinite P may or may not be finite. If P is finite then x(n) is called power signal. 8/5/2009 ٧ Energy Vs Power signals Notes: A signal is classified as energy type if its energy is finite ∞ (0<E< ) Energy signals have Zero power A signal is power type if its power is finite (0 < P < ∞) N 1 P = lim ∑ x[n] n →∞ 2 N + 1 n = − N Therefore, 8/5/2009 Prof. Dr Othman O Khalifa 2 1 EN N →∞ 2 N + 1 P = lim ٨ ٤ Periodic Vs Aperiodic signals – x(n+N) = x(n) for all n – If no value of N satisfy it the signal is aperiodic. A sinusoid x(n)= Asin2πfn is periodic if f=k/N Power of periodic signals = (1/N)∑0<n<N--1|x(n)|2 0<n<N 8/5/2009 ٩ Periodic Vs Aperiodic signals Notes: A periodic signal has no starting or finishing time. A periodic signal repeats endlessly. A signal that dos not repeats it self is siad to ne non-periodic or Aperiodic. 8/5/2009 Prof. Dr Othman O Khalifa ١٠ ٥ Symmetric (even) Vs Antisymmetric (odd) – x(n) = x(-n); symmetric (even) – x(n) = -x(-n); antisymmetric (odd) Any arbitrary signal can be expressed as sum of two signal components, even and odd. 8/5/2009 ١١ Examples 8/5/2009 Prof. Dr Othman O Khalifa ١٢ ٦ Classification of systems 1. 2. 3. Static (memoryless) Vs Dynamic (with memory) (memoryless) Time-invariant Vs Time Variant systems TimeLinear Vs Nonlinear (linear, must satisfy superposition principle, otherwise nonlinear.) nonlinear.) 4. Causal Vs Noncausal (o/p should depend upon present and past i/p but not on future i/p {x(n+1)..}, otherwise noncausal. If signal is first recorded then noncausal. offline processing is done then only non causal systems are possible to implement) 5 Stable Vs Unstable (BIBO stable if every bounded i/p produces a bounded o/p, o/p, otherwise unstable.) 8/5/2009 ١٣ Static (memoryless) Vs Dynamic (with memory) Memoryless System – A system is memoryless if the output y[n] at every value of n depends only on the depends input x[n] at the same value of n (static, if o/p depends upon present i/p only e.g. y(n)= nx(n)+bx3(n). y(n)= Dynamic, if depends upon past or present e.g. y(n)=x(n-1)+3x(n)) y(n)=x(nExample Memoryless Systems – Square y[n] = (x[n]) 2 – Sign Counter Example – Ideal Delay System y[n] = sign{x[n]} y[n] = x[n − no ] 8/5/2009 Prof. Dr Othman O Khalifa ١٤ ٧ Linear Vs Nonlinear Linear System: A system is linear if and only if T{x1[n] + x2[n]} = T{x1[n]} + T{x2[n]} (additivity) and T{ax[n]} = aT{x[n]} (scaling) Examples – Ideal Delay System y[n] = x[n − no ] T{x1[n] + x2[n]} = x1[n − no ] + x2[n − no ] T{x2 [n]} + T{x1[n]} = x1[n − no ] + x2[n − no ] T{ax[n]} = ax1[n − no ] aT{x[n]} = ax1[n − no ] 8/5/2009 ١٥ Time-invariant Vs Time Variant systems (if i/p, x(n) results in y(n) then x(n-k) must results in y(n-k), otherwise system i/p, x(n) y(n) x(ny(nis time variant. Ex: TI, y(n)=x(n)-x(n-1); TV, y(n)= x(n)*cos(wn)) y(n)=x(n)-x(ny(n)= x(n)*cos(wn)) y[n] = T{x[n]} ⇒ y[n − no ] = T{x[n − no ]} Time-Invariant (shift-invariant) Systems Time(shift- – A time shift at the input causes corresponding time-shift at output time- Delay the input the output is Example – Square 2 y[n] = (x[n]) Delay the output gives y1 [ n ] = ( x[n]) 2 y [ n-n o ] = ( x[ n − no ]) 2 Counter Example – Compressor System y[n] = x[Mn] 8/5/2009 Prof. Dr Othman O Khalifa Delay the input the output is Delay the output gives y1 [ n ] = x[ Mn] y [ n-n o ] = x ⎡ M ( n − no ) ⎤ ⎣ ⎦ ١٦ ٨ Causal Vs Noncausal Causality – A system is causal it’s output is a function of it’ only the current and previous samples Examples – Backward Difference Counter Example – Forward Difference y[n] = x[n] − x[n − 1] y[n] = x[n + 1] + x[n] 8/5/2009 ١٧ Stable Vs Unstable Stability (in the sense of bounded-input bounded-output BIBO) boundedbounded– A system is stable if and only if every bounded input produces a bounded output x[n] ≤ Bx < ∞ ⇒ y[n] ≤ By < ∞ 2 Example y[n] = (x[n]) – Square if input is bounded by x[n] ≤ B < ∞ x output is bounded by y[n] ≤ B2 < ∞ x Counter Example – Log y[n] = log10 ( x[n] ) even if input is bounded by x[n] ≤ Bx < ∞ output not bounded for x[n] = 0 ⇒ y[0] = log10 ( x[n] ) = −∞ 8/5/2009 Prof. Dr Othman O Khalifa ١٨ ٩ Discrete-Time Signals: Time-Domain Representation Signals represented as sequences of numbers, called samples Sample value of a typical signal or sequence denoted as x[n] with n being an integer in the range ∞ ≤ ≤ ∞ x[n] defined only for integer values of n and undefined for noninteger values of n Discrete-time signal represented by {x[n]} 8/5/2009 ١٩ Discrete-Time Signals: Sequences Discrete-time signals are represented by sequence of numbers Discrete– The nth number in the sequence is represented with x[n] Often times sequences are obtained by sampling of continuous-time signals continuous– In this case x[n] is value of the analog signal at xc(nT) (nT) – Where T is the sampling period 10 0 -10 0 10 t (ms) 20 40 60 80 100 10 20 30 40 50 n (samples) 0 -10 0 8/5/2009 Prof. Dr Othman O Khalifa ٢٠ ١٠ Basic Sequences and Operations y[n] = x[n − no ] 1.5 Delaying (Shifting) a sequence ⎧0 n ≠ 0 δ[n] = ⎨ ⎩1 n = 0 Unit sample (impulse) sequence ⎧0 n < 0 u[n] = ⎨ ⎩1 n ≥ 0 1 0.5 0 -10 0 5 10 -5 0 5 10 -5 0 5 10 ٢١ 1 0.5 0 -10 Unit step sequence -5 1.5 1 x[n] = Aαn Exponential sequences 0.5 0 -10 8/5/2009 Discrete-Time Systems Discrete-Time System is a mathematical operation that maps a given DiscreteSystem input sequence x[n] into an output sequence y[n] y[n] = T{x[n]} x[n] T{.} y[n] Example Discrete-Time Systems Discrete– Moving (Running) Average y[n] = x[n] + x[n − 1] + x[n − 2] + x[n − 3] – Maximum y[n] = max{x[n], x[n − 1], x[n − 2]} – Ideal Delay System y[n] = x[n − no ] 8/5/2009 Prof. Dr Othman O Khalifa ٢٢ ١١ Discrete-Time Signals: TimeDomain Representation 8/5/2009 ٢٣ 8/5/2009 ٢٤ Prof. Dr Othman O Khalifa ١٢ 8/5/2009 ٢٥ Discrete-Time Signals: Time-Domain Representation Discrete-Time Signals are represented mathematically as sequences Discreteof numbers. The sequences of numbers x with nth number in the sequence is denoted as x[n] where n being integer in the range of such as [n] n={…,-3,-2,-1,0,1,2,3,…}. It is also called time index. … 3,- 2,- 1,0,1,2,3,… n={ If the input signal to the systems is DTS, the output of the systems systems will be DTS. INPUT OUTPUT x[n] [n] 8/5/2009 Prof. Dr Othman O Khalifa SYSTEM y[n] [n] ٢٦ ١٣ Discrete-Time Signals: Time-Domain Representation Components of DTS ::=> A delay y[n] = x[n-1] [n] [n=> An advance y[n] = x[n+1] [n] => Scaling y[n] = ax[n] [n] [n] => Downsampling (Decimation) y[n] = x[nM] [n] [nM] => Upsampling (Interpolation) y[n] = x[n/L] [n] [n/L] => Nonlinear y[n] = tanh[ax(n)] [n] tanh[ax (n)] 8/5/2009 ٢٧ Discrete-Time Signals: Time-Domain Representation A DTS component is visualized below: 8/5/2009 Prof. Dr Othman O Khalifa ٢٨ ١٤ GRAPHICAL REPRESENTATION OF DTS Graphical Representation of a DTS is shown below: (infinite) 8/5/2009 ٢٩ Discrete-Time Signals: Time-Domain Representation The discrete-time signal is obtained by performing periodic discretesampling of analog signal. The sampling interval or period is denoted as Ts. Thus the sampling Frequency can be defined as reciprocal of Ts, namely, Fs = 1 / Ts. When the analog is sampled at certain period of time, the discrete-time signal can be written as below :discrete:- x[n] = xa[t] = xa[nTs], [n] [t] 8/5/2009 Prof. Dr Othman O Khalifa n = …,-2,-1,0,1,2,... 2,- ٣٠ ١٥ Discrete-Time Signals: Time-Domain Representation When the sinusoidal analog signal being sampled to become a discrete signal (digital signal), the frequency of input signal will be changed to digital frequency as illustrated below. Given a signal to the system is defined as: xa[t] = Acos(ωt + Ф) = Acos(2πFt + Ф) [t] Acos( Acos(2π The signal being sampled at certain period of time, Ts (Fs =1 / Ts). Thus the form of the discrete (digital signal) now become: xa[t] = x[n] = Acos(ωt + Ф) = Acos(2πFnT + Ф), t => nT, [t] [n] Acos( Acos(2π nT, x[n] = Acos(2πFnT + Ф) = Acos(2πnF/FS + Ф) [n] Acos(2π Acos(2π nF/F Hence, the digital Frequency is defined as : f = F/FS where F is Bandwidth input signal frequency & Fs is the Sampling Frequency and n is a time index. 8/5/2009 ٣١ Discrete-Time Signals: Time-Domain Representation Example : The input signal to the system is continuous cosine signal with Amplitude of 10 and signal frequency of 40Hz. The signal is sampled at sampled Sampling Frequency, Fs of 400Hz with sequence index, n ranging from 10 to 10. Steps : 1. Determine the Input Signal, xa[t] = 10cos(2*π*40*t) [t] 10cos(2*π since fo = 40Hz with zero phase shift (Ф = 0) 2. Calculate the Sampling Period, Ts = 1/Fs = 1/400 = 0.0025s 3. Determine the Discrete signal, x[n] = xa[t] = xa[nTs] = xa[n*0.0025] [n] [t] [n*0.0025] => 10cos(2*pi*40*n*0.0025), n = -10 to 10 EX. 4. Use MATLAB to compile the program and plot the graph 8/5/2009 Prof. Dr Othman O Khalifa ٣٢ ١٦ GRAPHICAL REPRESENTATION OF DISCRETEDISCRETETIME SIGNAL at TS = 0.0025s 8/5/2009 ٣٣ SEQUENCE REPRESENTATION There are several type of sequences in Discrete-Time Signals Discretesuch as unit sample or unit impulse, unit step, Real&complex exponential and Sinusoidal. The unit impulse is defined as : a. Unit Impulse => δ[n] = {1, n = 0; 0, n ≠ 0 } Graphical representation of Unit Impulse : 8/5/2009 Prof. Dr Othman O Khalifa ٣٤ ١٧ SEQUENCE REPRESENTATION b. Unit Step => u[n] = {1, n ≥ 0; [n] 0, n < 0 } Graphical representation of Unit Step : 8/5/2009 ٣٥ SEQUENCE REPRESENTATION c. 8/5/2009 Prof. Dr Othman O Khalifa Real exponential => x[n] = Aаn; A is constant and а is a [n] Aа real number. Graphical Representation of Real exponential : ٣٦ ١٨ SEQUENCE REPRESENTATION d. Complex exponential => x[n] = Aеjωn; ω frequency of [n] Aе complex exponential sinusoid, A is a constant Graphical representation of Complex exponential : 8/5/2009 ٣٧ SEQUENCE REPRESENTATION e. Sinusoidal 8/5/2009 Prof. Dr Othman O Khalifa ٣٨ ١٩ SEQUENCE REPRESENTATION e. The Unit Impulse can be shifted or delayed. The shifted Unit Impulse is Impulse denoted as ::δ[n-k] => The unit impulse is shifted to right by k δ[n+k] => The unit impulse is shifted to left by k n+k] Example : The sequence, p[n] is expressed as : p[n] p[n] = a-3 δ[n + 3] + a1 δ[n – 1] + a2 δ[n – 2] + a7 δ[n – 7] p[n] Graphical representation of Shifted unit Impulse of p[n]: p[n]: 8/5/2009 ٣٩ SEQUENCE REPRESENTATION The sequence of Discrete-Time Signals can be represented in Discreteterm of Shifted Unit impulse as defined below : ∞ x[n] = Σx[k]δ[n-k] [n] [k] k=-∞ k=- The unit step sequence can be defined in term of Shifted Unit Impulse as shown below : ∞ u[n] =Σδ[n-k] [n] =Σ k= 0 8/5/2009 Prof. Dr Othman O Khalifa ٤٠ ٢٠ SEQUENCE REPRESENTATION The unit impulse can be defined in term of Shifted Unit Step as shown below : δ[n] = u[n]-u[n-1] [n]- [n- 8/5/2009 ٤١ Introduction to LTI System Discrete-time Systems – Function: to process a given input sequence to generate an output sequence x[n] Input sequence Discrete-time system y[n] Output sequence Fig: Example of a single-input, single-output system 8/5/2009 Prof. Dr Othman O Khalifa ٤٢ ٢١ Review Classification of Discrete-time System System with and without memory Causality Stability Time-invariance Linearity 8/5/2009 ٤٣ Classification of Discrete-time System Memoryless system – Output at a given time is dependent on the input at only that same time y[n] = (2 x[n] − x 2 [n]) 2 Identity system y (t ) = x(t ) or y[n] = x[n] 8/5/2009 Prof. Dr Othman O Khalifa ٤٤ ٢٢ Classification of Discrete-time System System with memory – Summer n y[n] = ∑ x[k ] k =−∞ – Delay y[ n] = x[n − 1] 8/5/2009 ٤٥ Invertible – distinct inputs lead to distinct output – an inverse system exists Example y (t ) = 2 x(t ) → inverse system is w(t ) = 8/5/2009 Prof. Dr Othman O Khalifa 1 y (t ) 2 ٤٦ ٢٣ Classification of Discrete-time System Example y[n] = n ∑ x[k ] k =−∞ → inverse system is w[n] = y[n] - y[n -1] 8/5/2009 ٤٧ Classification of Discrete-time System 8/5/2009 Prof. Dr Othman O Khalifa ٤٨ ٢٤ Classification of Discrete-time System Causal – The output at any time depends on values of the input at only the present and past times – Example: memory system, RC circuit, …, etc. Noncausal y[n] = x[n] − x[n + 1] y (t ) = x(t + 1) 8/5/2009 ٤٩ Classification of Discrete-time System Example causal ⎧n > 0, (1) y[n] = x[−n] ⇒ ⎨ ⎩ n < 0, noncausal (2) y (t ) = x(t ) cos(t + 1) → y (t ) = x(t ) g (t ) where g (t ) = cos(t + 1) → causal system 8/5/2009 Prof. Dr Othman O Khalifa ٥٠ ٢٥ Classification of Discrete-time System Stable system – If the input is bounded then the output must be bounded and cannot diverge – Example: RC circuit … etc Unstable system – Example: y[n] = n ∑ u[k ] = (n + 1)u[n] k =−∞ 8/5/2009 ٥١ Classification of Discrete-time System Example – Look for a specific bounded input that leads to an unbounded output (1) S1 : y (t ) = tx(t ) <ans> if we consider x(t ) = 1 → y (t ) = t - - - -unstable 8/5/2009 Prof. Dr Othman O Khalifa ٥٢ ٢٦ Classification of Discrete-time System Example <ans> (2) S 2 : y (t ) = e x (t ) let B be an arbitrary positive number , and let x(t ) be an arbitrary signal bounded by B i.e. − B < x(t ) < B → e − B < y (t ) < e B − − − − stable 8/5/2009 ٥٣ Classification of Discrete-time System Time-invariant – The behavior and characteristics of the system are fixed over time – Example: RC circuit of Fig1.1 is timeinvariant if the resistance and capacitance values are constant – If a time-shift in the input signal results in an identical time shift in the output signal 8/5/2009 Prof. Dr Othman O Khalifa ٥٤ ٢٧ Classification of Discrete-time System Example y (t ) = sin[ x(t )] <ans> let y1 (t ) = sin[ x1 (t )] and x2 (t ) = x1 (t − t0 ) → y2 (t ) = sin[ x2 (t )] = sin[ x1 (t − t0 )] − −(a ) and y1 (t − t0 ) = sin[ x1 (t − t0 )] − − − − − (b) Q (a ) = (b) ∴ time − in var iance 8/5/2009 ٥٥ Classification of Discrete-time System Example y[n] = nx[n] <ans> let y1[n] = nx1[n] and x2 [n] = x1[n − n0 ] → y2 [n] = nx2 [n] = nx1[n − n0 ] − − − −(a ) and y1[n − n0 ] = (n − n0 ) x1[n − n0 ] − −(b) 8/5/2009 Prof. Dr Othman O Khalifa Q (a ) ≠ (b) ∴ time - var iance ٥٦ ٢٨ Classification of Discrete-time System Linear – Additivity property The response to x1 (t ) + x2 (t ) is y1 (t ) + y2 (t ) – Scaling (homogeneity) property The response to ax1 (t ) is ay1 (t ), where a is any complex cons tan t 8/5/2009 ٥٧ Classification of Discrete-time System Combine the two properties continuous : ax1 (t ) + bx2 (t ) → ay1 (t ) + by2 (t ) discrete : ax1[n] + bx2 [n] → ay1[n] + by2 [n] where a and b are complex cons tan ts Superposition property – An zero input results in an zero output 0 = 0 ⋅ x[n] → 0 ⋅ y[n] = 0 8/5/2009 Prof. Dr Othman O Khalifa ٥٨ ٢٩ Classification of Discrete-time System Example y (t ) = tx(t ) <ans> x1 (t ) → y1 (t ) = tx1 (t ) x2 (t ) → y2 (t ) = tx2 (t ) let x3 (t ) = ax1 (t ) + bx2 (t ), where a and b are arbitrary scalar → y3 (t ) = tx3 (t ) = t (ax1 (t ) + bx2 (t )) = atx1 (t ) + btx2 (t ) = ay1 (t ) + by2 (t ) → linear 8/5/2009 ٥٩ Classification of Discrete-time System Example y[n] = 2 x[n] + 3 <ans> if x[n] = 0 → y[n] = 3 ≠ 0 → nonlinear 8/5/2009 Prof. Dr Othman O Khalifa ٦٠ ٣٠ Classification of Discrete-time System Incrementally linear system – Difference between the responses to any two outputs to an incrementally linear system is a linear function of the difference between the two inputs – Example: y[n] = 2 x[n] + 3 → y1[n] − y2 [n] = {2 x1[n] + 3} − {2 x2 [ n] + 3} 8/5/2009 = 2{x1[n] − x2 [n]} ٦١ Classification of Discrete-time System Linear System – Most widely used – A Discrete-time system is a linear system if the Discretesuperposition principle always hold. – If y1[n] and y2[n] are the response to the input sequences x1[n] and x2[n], then x[n] = αx1[n] + βx2[n] 8/5/2009 Prof. Dr Othman O Khalifa Linear DTS y[n] = αy1[n] + βy2[n] ٦٢ ٣١ Classification of Discrete-time System Example Is the system described below linear or not ? y[n] = x[n] + x[n-1] [n] [n] [n- Step : a. Now, applying superposition by considering input as : x[n] = ax[n] + bx[n] [n] [n] [n] b. Substitute the equation above with equation in (a), become y[n] = (ax[n] + bx[n]) + (ax[n-1] + bx[n-1]) [n] (a [n] [n]) (ax [nbx [nc. Rearrange the equation above become ::y[n] = a(x[n] + x[n-1]) + b(x[n] + x[n-1]) => ay[n] + by[n] [n[n] [n] [n] a(x [n] [nb(x [n] c. The system is Linear since superposition is hold. 8/5/2009 ٦٣ Classification of Discrete-time System Shift-invariant System/Time-Invariant System – A shift (delay) in the input sequence cause a shift (shift) to the output sequence – If y1[n] is the response to an input x1[n], then the response to an input x[n] = x1[n - no] is y[n] = y1[n - no] 8/5/2009 Prof. Dr Othman O Khalifa ٦٤ ٣٢ Classification of Discrete-time System Causal System – Changes in output samples do not precede changes in input samples – y[no] depends only on x[n] for n ≤ no – Example: y[n] = x[n]-x[n-1] 8/5/2009 ٦٥ Classification of Discrete-time System Stable System – For every bounded input, the output is also bounded (BIBO) – Is the y[n] is the response to x[n], and if |x[n]| < Bx for all value of n then |y[n]| < By for all value of n Where Bx and By are finite positive constant 8/5/2009 Prof. Dr Othman O Khalifa ٦٦ ٣٣ Impulse and Step Response If the input to the DTS system is Unit Impulse (δ[n]), then output of the system will be Impulse Response (h[n]). If the input to the DTS system is Unit Step (μ[n]), then output of the system will be Step Response (s[n]). 8/5/2009 ٦٧ Input-output Relationship A Linear time-invariant system satisfied both the linearity and time invariance properties. An LTI discrete-time system is charact...
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