Discrete Signals Prof. Seungchul Lee Industrial AI Lab. Most - PowerPoint PPT Presentation

Discrete Signals Prof. Seungchul Lee Industrial AI Lab. Most slides from (edX) Discrete Time Signals and Systems by Prof. Richard G. Baraniuk 1

Discrete Time Signals • A signal 𝑦[𝑜] is a function that maps an independent variable to a dependent variable. • We will focus on discrete-time signals 𝑦[𝑜] – Independent variable is an integer: 𝑜 ∈ ℤ – Dependent variable is a real or complex number: 𝑦 𝑜 ∈ ℝ or ℂ 2

Plot Real Signals • Continuous signal 3

Plot Real Signals • Discrete signals 4

Discrete Signal Properties 5

Finite/Infinite Length Signals • An infinite-length discrete-time signal 𝑦[𝑜] is defined for all 𝑜 ∈ ℤ , i.e., −∞ < 𝑜 < ∞ • An finite-length discrete-time signal 𝑦[𝑜] is defined only for a finite range of 𝑂 1 ≤ 𝑜 ≤ 𝑂 2 6

Periodic Signals • A discrete-time signal is periodic if it repeats with period 𝑂 ∈ ℤ • The period 𝑂 must be an integer • A periodic signal is infinite in length 7

Periodic Signals • Convert a finite-length signal 𝑦[𝑜] defined for 𝑂 1 ≤ 𝑜 ≤ 𝑂 2 into an infinite-length signal by either – (infinite) zero padding, or – periodization with period 𝑂 8

Modular Arithmetic • Modular arithmetic with modulus 𝑂 takes place on a clock with 𝑂 – Modular arithmetic is inherently periodic • Periodization via Modular Arithmetic – Consider a length- 𝑂 signal 𝑦[𝑜] defined for 0 ≤ 𝑜 ≤ 𝑂 − 1 – A convenient way to express periodization with period 𝑂 is 𝑧 𝑜 = 𝑦 𝑜 𝑂 – Important interpretation • Infinite-length signals live on the (infinite) number line • Periodic signals live on a circle 9

Finite-Length and Periodic Signals • Finite-length and periodic signals are equivalent – All of the information in a periodic signal is contained in one period (of finite length) – Any finite-length signal can be periodized – Conclusion: We will think of finite-length signals and periodic signals interchangeably 10

Finite-Length and Periodic Signals • Shifting infinite-length signals – Given an infinite-length signal 𝑦[𝑜] , we can shift back and forth in time via 𝑦[𝑜 − 𝑛] – When 𝑛 > 0 , 𝑦[𝑜 − 𝑛] shifts to the right (forward in time, delay) – When 𝑛 < 0 , 𝑦[𝑜 − 𝑛] shifts to the left (back in time, advance) 11

Finite-Length and Periodic Signals • Shifting periodic signals – Periodic signals can also be shifted; consider 𝑧 𝑜 = 𝑦 (𝑜) 𝑂 – Shift one sample into the future: 𝑧 𝑜 − 1 = 𝑦 (𝑜 − 1) 𝑂 12

Shifting Finite-Length Signals • Consider finite-length signals 𝑦 and 𝑧 defined for 0 ≤ 𝑜 ≤ 𝑂 − 1 and suppose 𝑧 𝑜 = 𝑦[𝑜 − 1] • What to put in 𝑧[0] ? What to do with 𝑦[𝑂 − 1] ? We do not want to invent/lose information • Elegant solution: Assume 𝑦 and 𝑧 are both periodic with period 𝑂 ; then 𝑧 𝑜 = 𝑦 (𝑜 − 1) 𝑂 • This is called a periodic or circular shift 13

Circular Time Reversal • Example with 𝑂 = 8 14

Key Discrete Signals 15

Delta Function • Delta function = unit impulse = unit sample • The shifted delta function 𝜀[𝑜 − 𝑛] peaks up at 𝑜 = 𝑛 , (here 𝑛 = 5 ) 16

Delta Function Sample • Multiplying a signal by a shifted delta function picks out one sample of the signal and sets all other samples to zero • Important: 𝑛 is a fixed constant, and so 𝑦[𝑛] is a constant (and not a signal) 17

Unit Step Function • The shifted unit step 𝑣[𝑜 − 𝑛] jumps from 0 to 1 at 𝑜 = 𝑛 , (here 𝑛 = 4 ) 18

Unit Step Selects Part of a Signal • Multiplying a signal by a shifted unit step function zeros out its entries for 𝑜 < 𝑛 19

Real Exponential • The real exponential • For 𝑏 > 1 , 𝑦[𝑜] grows to the right • For 0 < 𝑏 < 1 , 𝑦[𝑜] shrinks to the right 20

Sinusoid Signals • There are two natural real-valued sinusoids: cos(𝜕𝑜 + 𝜚) and sin(𝜕𝑜 + 𝜚) – Frequency: 𝜕 (units: radians/sample) – Phase: 𝜚 (units: radians) – cos(𝜕𝑜) – sin(𝜕𝑜) 21

Sinusoid Signals • cos(0𝑜) 2𝜌 • cos 10 𝑜 4𝜌 • cos 10 𝑜 6𝜌 • cos 10 𝑜 10𝜌 • cos 10 𝑜 = cos 𝜌𝑜 22

Phase of Sinusoid 2𝜌 • cos 10 𝑜 2𝜌 𝜌 • cos 10 𝑜 − 2 2𝜌 2𝜌 • cos 10 𝑜 − 2 2𝜌 3𝜌 • cos 10 𝑜 − 2 2𝜌 4𝜌 2𝜌 • cos 10 𝑜 − = cos 10 𝑜 2 23

Complex Sinusoid 24

Complex Number • Adding 25

Euler’s Formula 26

Complex Number • Multiplying 27

Plot Complex Signals • When 𝑦 𝑜 ∈ ℂ , we can use two signal plots • For 𝑓 𝑘𝜕𝑜 phase 28

Geometrical Meaning of 𝒇 𝒋𝜾 • 𝑓 𝑗𝜄 : point on the unit circle with angle of 𝜄 • 𝜄 = 𝜕𝑢 • 𝑓 𝑗𝜕𝑢 : rotating on an unit circle with angular velocity of 𝜕 • Question: what is the physical meaning of 𝑓 −𝑗𝜕𝑢 ? 29

Sinusoidal Functions from Circular Motions 30

Sinusoidal Functions from Circular Motions 31

Discrete Sinusoids • Discrete Sinusoids 32

Visualize the Discrete Sinusoidals 33

Visualize the Discrete Sinusoidals 2𝜌 8 3, 𝑓 𝑘𝜕0 • 𝜕 = 34

Visualize the Discrete Sinusoidals 2𝜌 8 3, 𝑓 𝑘𝜕1 • 𝜕 = 35

Visualize the Discrete Sinusoidals 2𝜌 8 3, 𝑓 𝑘𝜕2 • 𝜕 = 36

Visualize the Discrete Sinusoidals 2𝜌 8 3, 𝑓 𝑘𝜕3 • 𝜕 = 37

Visualize the Discrete Sinusoidals 2𝜌 8 3, 𝑓 𝑘𝜕4 • 𝜕 = 38

Visualize the Discrete Sinusoidals 2𝜌 8 3, 𝑓 𝑘𝜕5 • 𝜕 = 39

Visualize the Discrete Sinusoidals 2𝜌 8 3, 𝑓 𝑘𝜕6 • 𝜕 = 40

Visualize the Discrete Sinusoidals 2𝜌 8 3, 𝑓 𝑘𝜕7 • 𝜕 = 41

Visualize the Discrete Sinusoidals 2𝜌 8 3, 𝑓 𝑘𝜕8 • 𝜕 = 42

Aliasing 43

Aliasing of Discrete Sinusoids • Consider two sinusoids with two different frequencies • But note that • The signal 𝑦 1 and 𝑦 2 have different frequencies but are identical • We say that 𝑦 1 and 𝑦 2 are aliases • This phenomenon is called aliasing 44

Alias-free Frequencies in Discrete Sinusoids • Alias-free frequencies – The only frequencies that lead to unique (distinct) sinusoids lie in an interval of length 2𝜌 – Two intervals are typically used in the signal processing 45

Low and High Frequencies in Discrete Sinusoids • Low frequencies: 𝜕 closed to 0 and 2𝜌 • High frequencies: 𝜕 closed to 𝜌 and −𝜌 2𝜌 • cos 20 𝑜 2𝜌 18𝜌 • cos 9 × 20 𝑜 = cos 20 𝑜 46

Low and High Frequencies in Discrete Sinusoids • Which one is a higher frequency? 47

Frequency in Discrete Sinusoids 48

Aliasing 49

Aliasing 50

Aliasing: Wheel 51

Visual Matrix of Discrete Sinusoids 52

Visual Matrix of Discrete Sinusoids 53

Complex Exponential Signals with Damping 54

Complex Exponential Signals • Consider the general complex number 𝑨 = 𝑨 𝑓 𝑘∠𝑨 , 𝑨 ∈ ℂ – 𝑨 = magnitude of 𝑨 – ∠𝑨 = phase angle of 𝑨 – Can visualize 𝑨 ∈ ℂ as a point in the complex plane • Complex exponential is a spiral – 𝑨 𝑜 is a real exponential envelope – 𝑓 𝑘𝜕𝑜 is a complex sinusoid – 𝑨 𝑜 is a helix 55

Damped Free Oscillation PHY245: Damped Mass On A Spring, https://www.youtube.com/watch?v=ZqedDWEAUN4 56

Plot Complex Signals • Rectangular form 57

Plot Complex Signals • Polar form 58

Signals are Vectors 59

Signals are Vectors • Vectors in ℝ 𝑂 or ℂ 𝑂 60

Transpose of a Vector • the transpose operation 𝑈 converts a column vector to a row vector (and vice versa) • In addition to transpose, the conjugate transpose (aka Hermitian transpose) operation 𝐼 takes the complex conjugate 61

Transpose in MATLAB • Be careful 62

Matrix Multiplication as Linear Combination • Linear Combination = Matrix Multiplication • Given a collection of 𝑁 vectors 𝑦 0, 𝑦 1, ⋯ 𝑦 𝑁−1 and scalars 𝛽 0, 𝛽 1, ⋯ 𝛽 𝑁−1 , the linear combination of the vectors is given by 63

Matrix Multiplication as Linear Combination • Step 1: stack the vectors 𝑦 𝑛 as column vectors into an 𝑂 × 𝑁 matrix • Step 2: stack the scalars 𝛽 𝑛 into an 𝑁 × 1 column vector • Step 3: we can now write a linear combination as the matrix/vector product • Note: the row- 𝑜 , column- 𝑛 element of the matrix 𝑌 𝑜,𝑛 = 𝑦 𝑛 [𝑜] 64

Inner Product • The inner product (or dot product) between two vectors 𝑦, 𝑧 ∈ ℂ 𝑂 is given by • The inner product takes two signals (vectors in ℂ 𝑂 ) and produces a single (complex) number • Inner product of a signal with itself • Two vectors 𝑦, 𝑧 ∈ ℂ 𝑂 are orthogonal if 65

Orthogonal Signals • Two sets of orthogonal signals 66