EE361: SIGNALS AND SYSTEMS II CH5: DISCRETE TIME FOURIER TRANSFORM - - PowerPoint PPT Presentation

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EE361: SIGNALS AND SYSTEMS II

CH5: DISCRETE TIME FOURIER TRANSFORM

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FOURIER TRANSFORM DERIVATION

CHAPTER 5.1-5.2 2

FOURIER SERIES REMINDER

 Previously, FS allowed representation of a periodic

signal as a linear combination of harmonically related exponentials

 𝑦[𝑜] = σ𝑙=<𝑂> 𝑏𝑙𝑓𝑘𝑙𝜕0𝑜

𝑏𝑙 = 1

𝑂 σ𝑜=<𝑂> 𝑦 𝑜 𝑓−𝑘𝑙𝜕0𝑜 𝑒𝑢

 𝜕0 =

2𝜌 𝑂

 Would like to extend this (Transform Analysis) idea

to aperiodic (non-periodic) signals

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DT FOURIER TRANSFORM DERIVATION

 Intuition (same idea as CTFT):  Consider a finite signal 𝑦[𝑜]  Periodic pad to get periodic signal ෤

𝑦[𝑜]

 Find FS representation of ෤

𝑦[𝑜]

 Analyze FS as 𝑂 → ∞ (𝜕0 → 0) to get DTFT

 Note DTFT is discrete in time domain – continuous in

frequency domain  Envelope 𝑌(𝑓𝑘𝜕) of normalized FS coefficients {𝑏𝑙𝑂}

defines the DTFT (spectrum of 𝑦[𝑜])

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DT FOURIER TRANSFORM PAIR

 𝑦[𝑜] =

1 2𝜌 ׬ 2𝜌 𝑌 𝑓𝑘𝜕 𝑓𝑘𝜕𝑜𝑒𝜕

synthesis eq (inverse FT)

 𝑌 𝑓𝑘𝜕 = σ𝑜=−∞

∞

𝑦 𝑜 𝑓−𝑘𝜕𝑜 analysis eq (FT)

 DTFT is discrete in time – continuous in frequency

 Notice the DTFT 𝑌(𝑓𝑘𝜕) is period with period 2𝜌

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DTFT CONVERGENCE

 The FT converges if

 σ𝑜 𝑦 𝑜

< ∞ absolutely summable

 σ𝑜 𝑦 𝑜

2 < ∞

finite energy

 iFT has not convergence issues because 𝑌 𝑓𝑘𝜕

is periodic

 Integral is over a finite 2𝜌 period (similar to FS)

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FT OF PERIODIC SIGNALS

 Important property

 𝑦 𝑜 = 𝑓𝑘𝑙𝜕0𝑜 ↔ 𝑌 𝑘𝜕 = σ𝑚=−∞

∞

2𝜌𝜀 𝜕 − 𝑙𝜕0 − 2𝜌𝑚

 Impulse at frequency 𝑙𝜕0 and 2𝜌 shifts

 Transform pair  σ𝑙=<𝑂> 𝑏𝑙𝑓𝑘𝑙𝜕0𝑜 ↔ 2𝜌 σ𝑙=−∞

∞

𝑏𝑙𝜀 𝜕 − 𝑙𝜕0

 Each 𝑏𝑙 coefficient gets turned into a delta at the

harmonic frequency

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DTFT PROPERTIES AND PAIRS

CHAPTER 5.3-5.6 8

PROPERTIES/PAIRS TABLES

 Most often will use Tables to solve problems  Table 5.1 pg 391 – DTFT Properties  Table 5.2 pg 392 – DTFT Transform Pairs

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NOTEWORTHY PROPERTIES

 Periodicity – 𝑌 𝑓𝑘𝜕 = 𝑌 𝑓𝑘 𝜕+2𝜌  Time shift – 𝑦 𝑜 − 𝑜0 ↔ 𝑓−𝑘𝜕𝑜0𝑌 𝑓𝑘𝜕  Frequency/phase shift – 𝑓𝑘𝜕0𝑜𝑦 𝑜 ↔ 𝑌 𝑓𝑘 𝜕−𝜕0  Convolution – 𝑦 𝑜 ∗ 𝑧 𝑜 ↔ 𝑌 𝑓𝑘𝜕 𝑍 𝑓𝑘𝜕  Multiplication – 𝑦 𝑜 𝑧 𝑜 ↔

1 2𝜌 ׬ 2𝜌 𝑌 𝑓𝑘𝜄 𝑍 𝑓𝑘 𝜕−𝜄

𝑒𝜄

 Notice this is an integral over a single period  periodic

convolution 1

2𝜌 𝑌 𝑓𝑘𝜕 ∗ 𝑍 𝑓𝑘𝜕

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NOTEWORTHY PAIRS I

 Decaying exponential

 ℎ 𝑜 = 𝑏𝑜𝑣[𝑜]

𝑏 < 1

 Magnitude

response

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 0 < 𝑏 < 1  Lowpass filter  −1 < 𝑏 < 0  Highpass filter

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DECAYING EXPONENTIAL

NOTEWORTHY PAIRS II

 Impulse

 𝑦 𝑜 = 𝜀[𝑜] ↔ 𝑌 𝑓𝑘𝜕 = σ𝑜 𝜀 𝑜 𝑓−𝑘𝜕𝑜 = σ𝑜 𝜀 𝑜 𝑓−𝑘𝜕 0 = σ𝑜 𝜀 𝑜 = 1  𝑦 𝑜 = 𝜀 𝑜 − 𝑜0 ↔ 𝑌 𝑓𝑘𝜕 = σ𝑜 𝜀 𝑜 − 𝑜0 𝑓−𝑘𝜕𝑜 = σ𝑜 𝜀 𝑜 − 𝑜0 𝑓−𝑘𝜕𝑜0 = 𝑓−𝑘𝜕𝑜0

 Rectangle pulse

 𝑦 𝑜 = ቊ1

𝑜 ≤ 𝑂1 𝑜 > 𝑂1 ↔ 𝑌 𝑓𝑘𝜕 = σ𝑜=−𝑂1

𝑂1

𝑓−𝑘𝜕𝑜 =

sin 𝜕 2𝑂1+1

2

sin 𝜕

2

 Periodic signal

 𝑦 𝑜 = σ𝑙=<𝑂> 𝑏𝑙𝑓𝑘𝑙𝜕0𝑜 ↔ 𝑌 𝑓𝑘𝜕 = 2𝜌 σ𝑙=−∞

∞

𝑏𝑙𝜀 𝜕 − 𝑙𝜕0

 One period of 𝑏𝑙 copied

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DTFT AND LTI SYSTEMS

CHAPTER 5.8 14

 Take FT of both sides  Solve for frequency response

 Rational form – ratio of

polynomials in e−𝑘𝜕

 Best solved using partial fraction

expansion (Appendix A)

 Note special heavy-side cover-up

approach for repeated root

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GENERAL DIFFERENCE EQUATION SYSTEM

LTI SYSTEM APPROACH

 Same techniques as in continuous case  𝑍 𝑓𝑘𝜕 = 𝐼 𝑓𝑘𝜕 𝑌 𝑓𝑘𝜕  Partial fraction expansion  Inverse FT with tables

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