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

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

Jan 12, 2023 235 likes β€’966 views

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 [] is a function that maps an independent variable to a

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Discrete Signals

  • Prof. Seungchul Lee

Industrial AI Lab.

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Most slides from (edX) Discrete Time Signals and Systems by Prof. Richard G. Baraniuk

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SLIDE 2

Discrete Time Signals

  • A signal 𝑦[π‘œ] is a function that maps an independent variable to a dependent variable.
  • We will focus on discrete-time signals 𝑦[π‘œ]

– Independent variable is an integer: π‘œ ∈ β„€ – Dependent variable is a real or complex number: 𝑦 π‘œ ∈ ℝ or β„‚

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  • Continuous signal

Plot Real Signals

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  • Discrete signals

Plot Real Signals

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Discrete Signal Properties

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Finite/Infinite Length Signals

  • An infinite-length discrete-time signal 𝑦[π‘œ] is defined for all π‘œ ∈ β„€, i.e., βˆ’βˆž < π‘œ < ∞
  • An finite-length discrete-time signal 𝑦[π‘œ] is defined only for a finite range of 𝑂1 ≀ π‘œ ≀ 𝑂2

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Periodic Signals

  • A discrete-time signal is periodic if it repeats with period 𝑂 ∈ β„€
  • The period 𝑂 must be an integer
  • A periodic signal is infinite in length

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SLIDE 8

Periodic Signals

  • Convert a finite-length signal 𝑦[π‘œ] defined for 𝑂1 ≀ π‘œ ≀ 𝑂2 into an infinite-length signal by either

– (infinite) zero padding, or – periodization with period 𝑂

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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[π‘œ] defined for 0 ≀ π‘œ ≀ 𝑂 βˆ’ 1 – A convenient way to express periodization with period 𝑂 is 𝑧 π‘œ = 𝑦 π‘œ 𝑂 – Important interpretation

  • Infinite-length signals live on the (infinite) number line
  • Periodic signals live on a circle

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SLIDE 10

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

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Finite-Length and Periodic Signals

  • Shifting infinite-length signals

– Given an infinite-length signal 𝑦[π‘œ], we can shift back and forth in time via 𝑦[π‘œ βˆ’ 𝑛] – When 𝑛 > 0, 𝑦[π‘œ βˆ’ 𝑛] shifts to the right (forward in time, delay) – When 𝑛 < 0, 𝑦[π‘œ βˆ’ 𝑛] shifts to the left (back in time, advance)

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Finite-Length and Periodic Signals

  • Shifting periodic signals

– Periodic signals can also be shifted; consider 𝑧 π‘œ = 𝑦 (π‘œ)𝑂 – Shift one sample into the future: 𝑧 π‘œ βˆ’ 1 = 𝑦 (π‘œ βˆ’ 1)𝑂

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SLIDE 13

Shifting Finite-Length Signals

  • Consider finite-length signals 𝑦 and 𝑧 defined for 0 ≀ π‘œ ≀ 𝑂 βˆ’ 1 and suppose 𝑧 π‘œ = 𝑦[π‘œ βˆ’ 1]
  • What to put in 𝑧[0]? What to do with 𝑦[𝑂 βˆ’ 1]? We do not want to invent/lose information
  • Elegant solution: Assume 𝑦 and 𝑧 are both periodic with period 𝑂; then 𝑧 π‘œ = 𝑦 (π‘œ βˆ’ 1)𝑂
  • This is called a periodic or circular shift

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Circular Time Reversal

  • Example with 𝑂 = 8

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Key Discrete Signals

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Delta Function

  • Delta function = unit impulse = unit sample
  • The shifted delta function πœ€[π‘œ βˆ’ 𝑛] peaks up at π‘œ = 𝑛, (here 𝑛 = 5)

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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)

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Unit Step Function

  • The shifted unit step 𝑣[π‘œ βˆ’ 𝑛] jumps from 0 to 1 at π‘œ = 𝑛, (here 𝑛 = 4)

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Unit Step Selects Part of a Signal

  • Multiplying a signal by a shifted unit step function zeros out its entries for π‘œ < 𝑛

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Real Exponential

  • The real exponential
  • For 𝑏 > 1, 𝑦[π‘œ] grows to the right
  • For 0 < 𝑏 < 1, 𝑦[π‘œ] shrinks to the right

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SLIDE 21

Sinusoid Signals

  • There are two natural real-valued sinusoids: cos(πœ•π‘œ + 𝜚) and sin(πœ•π‘œ + 𝜚)

– Frequency: πœ• (units: radians/sample) – Phase: 𝜚 (units: radians) – cos(πœ•π‘œ) – sin(πœ•π‘œ)

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Sinusoid Signals

  • cos(0π‘œ)
  • cos

2𝜌 10 π‘œ

  • cos

4𝜌 10 π‘œ

  • cos

6𝜌 10 π‘œ

  • cos

10𝜌 10 π‘œ = cos πœŒπ‘œ

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Phase of Sinusoid

  • cos

2𝜌 10 π‘œ

  • cos

2𝜌 10 π‘œ βˆ’ 𝜌 2

  • cos

2𝜌 10 π‘œ βˆ’ 2𝜌 2

  • cos

2𝜌 10 π‘œ βˆ’ 3𝜌 2

  • cos

2𝜌 10 π‘œ βˆ’ 4𝜌 2

= cos

2𝜌 10 π‘œ

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Complex Sinusoid

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Complex Number

  • Adding

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Euler’s Formula

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Complex Number

  • Multiplying

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Plot Complex Signals

  • When 𝑦 π‘œ ∈ β„‚, we can use two signal plots
  • For π‘“π‘˜πœ•π‘œ

phase

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Geometrical Meaning of π’‡π’‹πœΎ

  • π‘“π‘—πœ„: point on the unit circle with angle of πœ„
  • πœ„ = πœ•π‘’
  • π‘“π‘—πœ•π‘’: rotating on an unit circle with angular velocity of πœ•
  • Question: what is the physical meaning of π‘“βˆ’π‘—πœ•π‘’ ?

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Sinusoidal Functions from Circular Motions

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Sinusoidal Functions from Circular Motions

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Discrete Sinusoids

  • Discrete Sinusoids

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Visualize the Discrete Sinusoidals

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Visualize the Discrete Sinusoidals

  • πœ• =

2𝜌 8 3, π‘“π‘˜πœ•0

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SLIDE 35

Visualize the Discrete Sinusoidals

  • πœ• =

2𝜌 8 3, π‘“π‘˜πœ•1

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Visualize the Discrete Sinusoidals

  • πœ• =

2𝜌 8 3, π‘“π‘˜πœ•2

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SLIDE 37

Visualize the Discrete Sinusoidals

  • πœ• =

2𝜌 8 3, π‘“π‘˜πœ•3

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SLIDE 38

Visualize the Discrete Sinusoidals

  • πœ• =

2𝜌 8 3, π‘“π‘˜πœ•4

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SLIDE 39

Visualize the Discrete Sinusoidals

  • πœ• =

2𝜌 8 3, π‘“π‘˜πœ•5

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SLIDE 40

Visualize the Discrete Sinusoidals

  • πœ• =

2𝜌 8 3, π‘“π‘˜πœ•6

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SLIDE 41

Visualize the Discrete Sinusoidals

  • πœ• =

2𝜌 8 3, π‘“π‘˜πœ•7

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Visualize the Discrete Sinusoidals

  • πœ• =

2𝜌 8 3, π‘“π‘˜πœ•8

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Aliasing

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Aliasing of Discrete Sinusoids

  • Consider two sinusoids with two different frequencies
  • But note that
  • The signal 𝑦1 and 𝑦2 have different frequencies but are identical
  • We say that 𝑦1 and 𝑦2 are aliases
  • This phenomenon is called aliasing

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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

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Low and High Frequencies in Discrete Sinusoids

  • Low frequencies: πœ• closed to 0 and 2𝜌
  • High frequencies: πœ• closed to 𝜌 and βˆ’πœŒ
  • cos

2𝜌 20 π‘œ

  • cos 9 Γ—

2𝜌 20 π‘œ = cos 18𝜌 20 π‘œ

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Low and High Frequencies in Discrete Sinusoids

  • Which one is a higher frequency?

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Frequency in Discrete Sinusoids

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Aliasing

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Aliasing

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Aliasing: Wheel

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Visual Matrix of Discrete Sinusoids

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Visual Matrix of Discrete Sinusoids

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Complex Exponential Signals with Damping

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Complex Exponential Signals

  • Consider the general complex number 𝑨 = 𝑨 π‘“π‘˜βˆ π‘¨, 𝑨 ∈ β„‚

– 𝑨 = magnitude of 𝑨 – βˆ π‘¨ = phase angle of 𝑨 – Can visualize 𝑨 ∈ β„‚ as a point in the complex plane

  • Complex exponential is a spiral

– 𝑨 π‘œ is a real exponential envelope – π‘“π‘˜πœ•π‘œ is a complex sinusoid – π‘¨π‘œ is a helix

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Damped Free Oscillation

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

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Plot Complex Signals

  • Rectangular form

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Plot Complex Signals

  • Polar form

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Signals are Vectors

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Signals are Vectors

  • Vectors in ℝ𝑂 or ℂ𝑂

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Transpose of a Vector

  • the transpose operation π‘ˆ 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

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Transpose in MATLAB

  • Be careful

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Matrix Multiplication as Linear Combination

  • Linear Combination = Matrix Multiplication
  • Given a collection of 𝑁 vectors 𝑦0,𝑦1, β‹― π‘¦π‘βˆ’1 and scalars 𝛽0,𝛽1, β‹― π›½π‘βˆ’1, the linear combination of the

vectors is given by

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Matrix Multiplication as Linear Combination

  • Step 1: stack the vectors 𝑦𝑛 as column vectors into an 𝑂 Γ— 𝑁 matrix
  • Step 2: stack the scalars 𝛽𝑛 into an 𝑁 Γ— 1 column vector
  • Step 3: we can now write a linear combination as the matrix/vector product
  • Note: the row-π‘œ , column-𝑛 element of the matrix π‘Œ π‘œ,𝑛 = 𝑦𝑛[π‘œ]

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Inner Product

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

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Orthogonal Signals

  • Two sets of orthogonal signals

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Harmonic Sinusoids are Orthogonal

  • Claim: 𝑒𝑙, π‘’π‘š = 0, 𝑙 β‰  π‘š

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Harmonic Sinusoids are Orthogonal

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Harmonic Sinusoids are Orthogonal

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Normalized Harmonic Sinusoids

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Matrix Multiplication as a Sequence of Inner Products of Rows

  • Consider the matrix multiplication 𝑧 = π‘Œπ›½
  • The row-π‘œ , column-𝑛 element of the matrix π‘Œ π‘œ,𝑛 = 𝑦𝑛[π‘œ]
  • We can compute each element 𝑧[π‘œ] in 𝑧 as the inner product of the π‘œ-th row of π‘Œ with the vector 𝛽
  • Can write 𝑧[π‘œ]

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