CEN352 Dr. Nassim Ammour King Saud University Chapter 3 Digital Signals and Systems CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Introduction This chapter introduces notations for digital signals and special digital sequences. The chapter continues to study some properties of linear systems such as time invariance, causality, impulse response, difference equations, and digital convolution. CEN352 Dr. Nassim Ammour King Saud University

Graphical representation of a discrete-time signal 𝒙(𝒏) Digital Signals A discrete-time signal 𝑥(𝑛) is a function of an integer variable 𝑛, where 𝑛∈ℤ. Graphical representation of a discrete-time signal 𝒙(𝒏) For floating point DS processor, the amplitudes can be floating points. CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Digital Signals Contd. Besides the graphical representation, there are some alternative representations. 𝑥 𝑛 = 1, 𝑓𝑜𝑟 𝑛=1,3 4, 𝑓𝑜𝑟 𝑛=2 0, 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒 1. Functional representation 2. Tabular representation 3. Sequence representation (bold or arrow for origin n=0) 𝑥 𝑛 = …,0 ,𝟎 ,1 ,4 ,1, 0, 0,… 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒−𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑠𝑖𝑔𝑛𝑎𝑙 𝑥 𝑛 = 0 ,−2 ,1 ,4 ,−1, 𝑓𝑖𝑛𝑖𝑡𝑒−𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑠𝑖𝑔𝑛𝑎𝑙 CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Common Digital Sequences 1. Unit-impulse sequence: 2. Unit-step sequence: CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Shifted Sequences CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Sinusoidal and Exponential Sequences CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Example 1 The sum CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Generation of Digital Signals To generate the digital sequence 𝑥 𝑛 from the analog signal 𝑥 𝑡 : uniformly sampling at the time interval of ∆𝑡=𝑇 𝑥 𝑛 = 𝑥(𝑡) 𝑡=𝑛𝑇 =𝑥(𝑛𝑇) Example 2 Convert analog signal 𝑥 𝑡 into digital signal 𝑥 𝑛 , when sampling period is 125 microsecond, also plot sample values. 𝑥 𝑡 =10 𝑒 −5000𝑡 𝑢(𝑡 Solution: CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Example 2 (contd.) CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Classification of Discrete-Time Signals 𝐸= 𝑛=−∞ ∞ 𝑥 𝑛 2 Energy signals: The energy E of a signal 𝑥 𝑛 Periodic signals and aperiodic signals: A signal 𝑥 𝑛 is periodic with period N (N>0) if 𝑥 𝑛+𝑁 =𝑥 𝑛 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑛 Symmetric (even) and anti-symmetric (odd) signals Anti-symmetric 𝑥 −𝑛 =−𝑥 𝑛 symmetric 𝑥 −𝑛 =𝑥 𝑛 CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Digital Systems A digital system is a device or an algorithm that performs prescribed operations (or transformation) on a digital signal 𝑥 𝑛 and produces an output signal 𝑦 𝑛 . Digital System Input signal (Excitation) Output signal (Response) 𝑥 𝑛 𝑦 𝑛 Example 3 𝑥 𝑛 = 𝑛 , −3≤𝑛≤3 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Determine the response of the following system to the input signal 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑠𝑦𝑠𝑡𝑒 𝑚 ′ 𝑠 𝑜𝑢𝑡𝑝𝑢𝑡 𝑦 𝑛 = 1 3 𝑥 𝑛+1 +𝑥 𝑛 +𝑥(𝑛−1 Solution: The output of this system is the mean value of the present, immediate past, and the immediate future samples. For 𝑛=0 𝑦 0 = 1 3 𝑥 −1 +𝑥 0 +𝑥(1) = 1 3 1+0+1 = 2 3 𝑦 𝑛 = …, 0,1, 5 3 , 2, 1, 2 3 , 1, 2, 5 3 , 1, 0, … Repeating this computation for every value of 𝑛 CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Block Diagram of Discrete-Time Systems Constant multiplier Unit delay element Adder Unit advance element Signal multiplier Example 4 Solution: Sketch the block diagram representation of the discrete-time system described by the input-output relation. 𝑦 𝑛 = 1 4 𝑦 𝑛−1 + 1 2 𝑥 𝑛 + 1 2 𝑥(𝑛−1) CEN352 Dr. Nassim Ammour King Saud University

Classification of Discrete-Time Systems Static and dynamic systems 𝑦 𝑛 =𝑥 𝑛 +3𝑥(𝑛−1 𝑦 𝑛 =𝑛 𝑥 𝑛 +𝑏 𝑥 3 (𝑛 Dynamic (have memory) past or future samples of the input Static or memory-less no past or future samples of the input Time-invariant System (Time Shift Invariance) The system input-output characteristics do not change with time (a shifted input signal will produce a shifted output signal, with the same of shifting amount). CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Linear System A digital system is linear if and only if it satisfy the superposition principle: 𝑆 𝑎 1 𝑥 1 𝑛 + 𝑎 2 𝑥 2 𝑛 = 𝑎 1 𝑆 𝑥 1 𝑛 + 𝑎 2 𝑆 𝑥 2 𝑛 Homogeneity (deals with amplitude) Additivity Homogeneity & Additivity CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Example 5 (a) CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Example 5 (b) System 𝐿𝑖𝑛𝑒𝑎𝑟 ??? 4 𝑦 1 𝑛 +2 𝑦 2 𝑛 𝑦(𝑛)≠4 𝑦 1 𝑛 +2 𝑦 2 𝑛 CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Example 6 (a) Given the linear system y 𝑛 =2𝑥(𝑛−5), find whether the system is time invariant or not. Solution: Let the shifted input be: 𝑥 2 𝑛 = 𝑥 1 (𝑛− 𝑛 0 ), Therefore system output: 𝑦 2 𝑛 =2 𝑥 2 𝑛−5 =2 𝑥 1 𝑛− 𝑛 0 −5 , Shifting 𝑦 1 𝑛 =2 𝑥 1 𝑛−5 by 𝑛 0 samples leads to 𝑦 1 𝑛− 𝑛 0 =2 𝑥 1 𝑛−5− 𝑛 0 CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Example 6 (b) Given the linear system y 𝑛 =𝑛 𝑥(𝑛), find whether the system is time invariant or not. 𝑦 1 𝑛 =𝑛 𝑥 1 (𝑛) Solution: Let the shifted input be: 𝑥 2 𝑛 = 𝑥 1 (𝑛− 𝑛 0 ) Therefore system output due to the shifted input 𝑦 2 𝑛 =𝑛 𝑥 2 𝑛 =𝑛 𝑥 1 𝑛− 𝑛 0 Shifting 𝑦 1 𝑛 =𝑛 𝑥 1 𝑛 by 𝑛 0 samples (replace 𝑛 by 𝑛−𝑛 0 ) leads to 𝑦 1 𝑛− 𝑛 0 =(𝑛− 𝑛 0 ) 𝑥 1 𝑛− 𝑛 0 CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Causality Causal System: Output y(n) at time n depends only on current input 𝑥 𝑛 , at time n or previous inputs, such as 𝑥 𝑛 , 𝑥 𝑛−1 , 𝑥 𝑛−2 , etc. Example: Non Causal System: Output y(n) at time n depends on future inputs, such as 𝑥 𝑛+1 , 𝑥 𝑛+2 , etc. Example: The non causal system cannot be realized in real time. CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Difference Equation A causal, linear, time-invariant system (LTI) can be described by a difference equation as follow: Outputs Inputs After rearranging: Finally: CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Example 7 Identify non zero system coefficients of the following difference equations. Solution: CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University System Representation Using Impulse Response A linear time-invariant system (LTI system) can be completely described by its unit-impulse response ℎ(𝑛) due to the impulse input 𝛿(𝑛) with zero initial conditions. Impulse input with zero initial conditions Impulse Response 𝑦 𝑛 =ℎ 𝑛 ⊗𝑥 𝑛 Any input Convolution CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Example 8 (a) Given the linear time-invariant system: Solution: a. let 𝑥 𝑛 =𝛿(𝑛), then Therefore, c. The system output b. The block diagram of the LTI system CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Example 8 (b) Solution: a. Let 𝑥 𝑛 =𝛿(𝑛), then With the calculated results, we can predict the impulse response as CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Example 8 (b) – contd. b. The system block diagram c. The output sequence Finite Impulse Response (FIR) system: When the difference equation contains no previous outputs, i.e. ‘a’ coefficients are zero. ( See example 8 (a) ) Infinite Impulse Response (IIR) system: When the difference equation contains previous outputs, i.e. ‘a’ coefficients are not all zero. ( See example 8 (b) ) CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University BIBO Stability BIBO: Bounded In and Bounded Out A stable system is one for which every bounded input produces a bounded output. We have: Let, in the worst case, every input value reaches to maximum value M. Using absolute values of the impulse responses, If the impulse responses are finite number, then output is also finite. CEN352 Dr. Nassim Ammour King Saud University Stable system.

CEN352 Dr. Nassim Ammour King Saud University BIBO Stability –contd. To determine whether a system is stable, we apply the following equation: Impulse response is decreasing to zero. CEN352 Dr. Nassim Ammour King Saud University

The summation is finite, so the system is stable. Example 9 Given a linear system given by: Which is described by the unit-impulse response: Determine whether the system is stable or not. Solution: Using definition of step function: For a< 1, we know The summation is finite, so the system is stable. Therefore CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Digital Convolution A LTI system can be represented using a digital convolution sum. The unit-impulse response ℎ(𝑛) relates the system input and output. Commutative The sequences are interchangeable. Convolution sum requires ℎ(𝑛) to be reversed and shifted. If ℎ(𝑛) is the given sequence, ℎ(−𝑛) is the reversed sequence. CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Reversed Sequence Given a sequence Sketch the sequence ℎ(𝑘) and reversed sequence ℎ(−𝑘). Solution: CEN352 Dr. Nassim Ammour King Saud University

Convolution Using Table Method Example 9 Solution: Length =3 Length =3 CEN352 Dr. Nassim Ammour King Saud University Convolution length = 3 +3 –1 = 5

Convolution Using Table Method Example 10 Solution: Convolution length = 3 + 2 –1 = 4 CEN352 Dr. Nassim Ammour King Saud University

CEN352 Dr. Nassim Ammour King Saud University Convolution Properties CEN352 Dr. Nassim Ammour King Saud University