6.003: Signal Processing

Transcription

1 6.003: Signal Processing Coninuous-Time Fourier Transform Definiion Examples Properies Relaion o Fourier Series Sepember 5, 08

2 Quiz Thursday, Ocober 4, from 3pm o 5pm. No lecure on Ocober 4. The exam is closed book. No elecronic devices. You may use one 8.5x shee of noes (fron and back). Coverage: lecures, labs, reciaions, and homeworks up o and including Ocober 3. Pracice problems will be posed by he end of his week.

3 From Fourier Series o Fourier Transforms Las week we represened periodic signals as sums of sinusoids. This represenaion provides insighs ha are no obvious from oher represenaions (e.g., as funcions of ime). consonance and dissonance (lecure ) change in pich of a siren (lab/reciaion 3b) However, here are limiaions. only works for periodic signals (lecure 3a) mus know signal s period before doing he analysis (pse 3 #7) Today: avoid hese limiaion by defining he Fourier ransform.

4 From Fourier Series o Fourier Transforms How can we represen an aperiodic signal as a sum of sinusoids? x() S 0 S Sraegy: make a periodic version of x() by summing shifed copies: x p () = x( it ) i= x p () T S 0 S T Since x p () is periodic, i has a Fourier series (which depends on T ). Take he limi as T. As x p () x(), he Fourier series will approach he Fourier ransform.

5 From Fourier Series o Fourier Transforms Example: x p () T S 0 S T Calculae he Fourier series coefficiens X p [k]: X p [k] = j π x p ()e T k sin πk d = T S T T T πk T Express X p [k] in erms of = πk T : X p [k] = sin S T T X() As T, T/ T X p [k] = x p ()e j d X() = x()e j d T/ X() is he Fourier ransform of x().

6 Relaion Beween Fourier Transform and Fourier Series Fourier series coefficiens are samples of coninuous funcion of freq. sin πks T X p [k] = T sin S = = X() πk T If S = : 4 X() = π T k π 0 π = π T k If T = 8 and S = : X() = T X p [k] 4 π 0 π If T = 6 and S = : X() = T X p [k] 4 π 0 π = π T k = π T k

7 Relaion Beween Fourier Transform and Fourier Series We can reconsuc x() from X() using Riemann sums. x p () = X p [k]e j π T k = T X p [k]e j π T k ( π π T ) X()e j d π k k S X() π 0 π S = ; T = 8 X() = T X p [k] 4 π 0 π S = ; T = 6 X() = T X p [k] 4 π 0 π π T π T = π T k = π T k = π T k Fourier Transform relaion: x() f X()

8 Relaion Beween Fourier Transform and Fourier Series Fourier series / ransforms express signals by heir frequency conen. Coninuous-Time Fourier Series X[k] = x()e jko d T T x() = x( + T ) = X[k]e jko k= Coninuous-Time Fourier Transform X()= x()e j d x() = X()e j d π analysis equaion synhesis equaion where o = π T analysis equaion synhesis equaion

9 Examples of Fourier Transforms Find he Fourier Transform (FT) of a recangular pulse: { < < x() = 0 oherwise x() 0 X() = x()e j d = e j d = e j j X() 4π π 0 π 4π = sin The FT is a recipe for consrucing x() from sinusoidal componens: x() = X()e j d π A square pulse conains almos all frequencies.

10 Examples of Fourier Transforms The Fourier ransform of a recangular pulse is sin. x() f 0 X() 4π π 0 π 4π X() conains all frequencies excep non-zero muliples of π. Why is he ransform zero a non-zero muliples of π? Wha is special abou hose frequencies? Why isn i zero a = 0?

11 Check Yourself There are n periods of e jnπ = cos nπ j sin nπ in < <. x() 0 cos π sin π { X( = nπ) = e jnπ if n = 0 d = 0 oherwise No Fourier componens are needed for frequencies = nπ. However DC is required o offse x() so ha x() 0 for all.

12 Examples of Fourier Transforms Find he Fourier Transform of a delayed recangular pulse: { 0 < < x d () = 0 oherwise x d () X d () = x d ()e j d = 0 0 e j d = e j j = j e j( e j e j) j sin = e 0 = e j X() = ( e j ) j

13 Properies of Fourier Transforms Time delays map o linear phase delay of he Fourier ransform. If x() hen x( τ) f f X() e jτ X() X() = x()e j d Y () = x( τ)e j d Le u = τ (and herefore du = d since τ is a consan) Y () = x(u)e j(u+τ) du = e jτ x(u)e ju du = e jτ X()

14 Examples of Fourier Transforms Time delay. x() 0 f X() X() 4π 4π π 4π π 4π X d () x d () 0 f X d () 4π 4π π π 4π 4π Jus enough phase o delay every frequency componen by =.

15 Fourier Transform Scaling ime. Consider he following signal and is Fourier ransform. Time represenaion: x () Frequency represenaion: X () = sin π How would hese scale if ime were sreched?

16 Check Yourself Signal x () and is Fourier ransform X () are shown below. x () X () b 0 Which of he following is rue?. b = and 0 = π/. b = and 0 = π 3. b = 4 and 0 = π/ 4. b = 4 and 0 = π 5. none of he above

17 Check Yourself Find he Fourier ransform. X () = e j d = e j j = sin = 4 sin 4 π/

18 Check Yourself Signal x () and is Fourier ransform X () are shown below. x () X () b 0 Which of he following is rue? 3. b = and 0 = π/. b = and 0 = π 3. b = 4 and 0 = π/ 4. b = 4 and 0 = π 5. none of he above

19 Fourier Transforms Sreching ime compresses frequency. x () X () = sin π X () = 4 sin x () 4 π/

20 Fourier Transforms Find a general scaling rule. Le x () = x (a). X () = x ()e j d = x (a)e j d Le τ = a (a > 0, i.e., no flipped in ime). X () = x (τ)e jτ/a a dτ = ( ) a X a If a < 0 he sign of dτ would change along wih he limis of inegraion. In general, x (a) ( ) a X a. If ime is sreched (0 < a < ) hen frequency is compressed and ampliude increases (preserving area).

21 Momens The value of X() a = 0 is he inegral of x() over ime. X() =0 = x()e j d = x()e j0 d = x() d x () X () = sin area = π

22 Momens The value of x(0) is he inegral of X() divided by π. x(0) = X() e j d = X() d π π x () X () = sin area π = π

23 Momens The value of x(0) is he inegral of X() divided by π. x(0) = X() e j d = X() d π π x () X () = sin area π = π equal areas! π

24 Sreching Time Sreching ime compresses frequency and increases ampliude (preserving area). x () X () = sin π 4 π

25 Compressing Time o he Limi Alernaively, we could compress ime while keeping area =. x() X() = sin / / π 4 4 π In he limi, he pulse has zero widh bu area! We represen his limi wih he dela funcion: δ().

26 Summary: Coninuous-Time Fourier Transform (CTFT) Definiion analysis and synhesis relaions: analogous o CTFS Examples square pulse Properies ime delay ime scaling momen relaions Relaion o Fourier Series ( X() = π X[k] δ k π ) T k=

27 Today s Lab and Reciaion Pracice wih Fourier Transforms