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

1 EE361: SIGNALS AND SYSTEMS II CH5: DISCRETE TIME FOURIER TRANSFORM http://www.ee.unlv.edu/~b1morris/ee361

2 FOURIER TRANSFORM DERIVATION CHAPTER 5.1-5.2

3 FOURIER SERIES REMINDER  Previously, FS allowed representation of a periodic signal as a linear combination of harmonically related exponentials 𝑏 𝑙 = 1 𝑂 σ 𝑜=<𝑂> 𝑦 𝑜 𝑓 −𝑘𝑙𝜕 0 𝑜 𝑒𝑢  𝑦[𝑜] = σ 𝑙=<𝑂> 𝑏 𝑙 𝑓 𝑘𝑙𝜕 0 𝑜 2𝜌  𝜕 0 = 𝑂  Would like to extend this (Transform Analysis) idea to aperiodic (non-periodic) signals

4 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(𝑓 𝑘𝜕 ) of normalized FS coefficients {𝑏 𝑙 𝑂} defines the DTFT (spectrum of 𝑦[𝑜] )

5 DT FOURIER TRANSFORM PAIR 1 2𝜌 𝑌 𝑓 𝑘𝜕 𝑓 𝑘𝜕𝑜 𝑒𝜕 synthesis eq (inverse FT)  𝑦[𝑜] = 2𝜌 ׬  𝑌 𝑓 𝑘𝜕 = σ 𝑜=−∞ ∞ 𝑦 𝑜 𝑓 −𝑘𝜕𝑜 analysis eq (FT)  DTFT is discrete in time – continuous in frequency  Notice the DTFT 𝑌(𝑓 𝑘𝜕 ) is period with period 2𝜌

6 DTFT CONVERGENCE  The FT converges if absolutely summable  σ 𝑜 𝑦 𝑜 < ∞ 2 < ∞ finite energy  σ 𝑜 𝑦 𝑜  iFT has not convergence issues because 𝑌 𝑓 𝑘𝜕 is periodic  Integral is over a finite 2𝜌 period (similar to FS)

7 FT OF PERIODIC SIGNALS  Important property  𝑦 𝑜 = 𝑓 𝑘𝑙𝜕 0 𝑜 ↔ 𝑌 𝑘𝜕 = σ 𝑚=−∞ ∞ 2𝜌𝜀 𝜕 − 𝑙𝜕 0 − 2𝜌𝑚  Impulse at frequency 𝑙𝜕 0 and 2𝜌 shifts  Transform pair  σ 𝑙=<𝑂> 𝑏 𝑙 𝑓 𝑘𝑙𝜕 0 𝑜 ↔ 2𝜌 σ 𝑙=−∞ ∞ 𝑏 𝑙 𝜀 𝜕 − 𝑙𝜕 0  Each 𝑏 𝑙 coefficient gets turned into a delta at the harmonic frequency

8 DTFT PROPERTIES AND PAIRS CHAPTER 5.3-5.6

9 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

10 NOTEWORTHY PROPERTIES  Periodicity – 𝑌 𝑓 𝑘𝜕 = 𝑌 𝑓 𝑘 𝜕+2𝜌  Time shift – 𝑦 𝑜 − 𝑜 0 ↔ 𝑓 −𝑘𝜕𝑜 0 𝑌 𝑓 𝑘𝜕  Frequency/phase shift – 𝑓 𝑘𝜕 0 𝑜 𝑦 𝑜 ↔ 𝑌 𝑓 𝑘 𝜕−𝜕 0  Convolution – 𝑦 𝑜 ∗ 𝑧 𝑜 ↔ 𝑌 𝑓 𝑘𝜕 𝑍 𝑓 𝑘𝜕 1 2𝜌 𝑌 𝑓 𝑘𝜄 𝑍 𝑓 𝑘 𝜕−𝜄  Multiplication – 𝑦 𝑜 𝑧 𝑜 ↔ 𝑒𝜄 2𝜌 ׬  Notice this is an integral over a single period  periodic convolution 1 2𝜌 𝑌 𝑓 𝑘𝜕 ∗ 𝑍 𝑓 𝑘𝜕

11 NOTEWORTHY PAIRS I  Decaying exponential  ℎ 𝑜 = 𝑏 𝑜 𝑣[𝑜] 𝑏 < 1  Magnitude response

12 DECAYING EXPONENTIAL  0 < 𝑏 < 1  −1 < 𝑏 < 0  Lowpass filter  Highpass filter

13 NOTEWORTHY PAIRS II  Impulse  𝑦 𝑜 = 𝜀[𝑜] ↔ 𝑌 𝑓 𝑘𝜕 = σ 𝑜 𝜀 𝑜 𝑓 −𝑘𝜕𝑜 = σ 𝑜 𝜀 𝑜 𝑓 −𝑘𝜕 0 = σ 𝑜 𝜀 𝑜 = 1  𝑦 𝑜 = 𝜀 𝑜 − 𝑜 0 ↔ 𝑌 𝑓 𝑘𝜕 = σ 𝑜 𝜀 𝑜 − 𝑜 0 𝑓 −𝑘𝜕𝑜 = σ 𝑜 𝜀 𝑜 − 𝑜 0 𝑓 −𝑘𝜕𝑜 0 = 𝑓 −𝑘𝜕𝑜 0  Rectangle pulse sin 𝜕 2𝑂1+1  𝑦 𝑜 = ቊ1 𝑜 ≤ 𝑂 1 ↔ 𝑌 𝑓 𝑘𝜕 = σ 𝑜=−𝑂 1 𝑓 −𝑘𝜕𝑜 = 𝑂 1 2 sin 𝜕 0 𝑜 > 𝑂 1 2  Periodic signal  𝑦 𝑜 = σ 𝑙=<𝑂> 𝑏 𝑙 𝑓 𝑘𝑙𝜕 0 𝑜 ↔ 𝑌 𝑓 𝑘𝜕 = 2𝜌 σ 𝑙=−∞ ∞ 𝑏 𝑙 𝜀 𝜕 − 𝑙𝜕 0  One period of 𝑏 𝑙 copied

14 DTFT AND LTI SYSTEMS CHAPTER 5.8

15 GENERAL DIFFERENCE EQUATION SYSTEM  Solve for frequency response  Take FT of both sides  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

16 LTI SYSTEM APPROACH  Same techniques as in continuous case  𝑍 𝑓 𝑘𝜕 = 𝐼 𝑓 𝑘𝜕 𝑌 𝑓 𝑘𝜕  Partial fraction expansion  Inverse FT with tables