Young Won Lim4/20/13

● Discrete Time Rect Functions

Discrete Time Rect Function(4B)

Young Won Lim4/20/13

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3 Young Won Lim4/20/13DT Rect (4B)

Fourier Transform Types

Discrete Time Fourier Transform

X e j = ∑n=−∞

∞

x [n ] e− j n x [n] = 12π ∫−π

+πX (e j ω̂) e+ j ω̂n

Discrete Fourier Transform

X [k ] = ∑n= 0

N − 1

x [n] e− j2 /N k n x [n ] = 1N ∑

k = 0

N − 1

X [k ] e j 2/N k n

4 Young Won Lim4/20/13DT Rect (4B)

DTFT and DTFS

DTFS (Discrete Time Fourier Series)

DTFT (Discrete Time Fourier Transform)X (e j ω̂) = sin(ω̂ L/2)

sin(ω̂/2) = L DL(ej ω̂)

X [k ] = 1N 0

sin (π k L /N 0)sin(π k /N 0)

= LN 0

⋅drcl (k /N 0 , L)

N 0

(L−1) zerocrossings

L/N 0

0

1

+N−N

L = 2 N+1 N 0

2πL

= L⋅diric(ω̂ , L)

(L−1) zerocrossings

0

1

+N−N

L = 2 N+1

5 Young Won Lim4/20/13DT Rect (4B)

6 Young Won Lim4/20/13DT Rect (4B)

RectN[n] DTFT

0

1

+N−N

Discrete Time Fourier Transform DTFT

X e j = ∑n=−∞

∞

x [n ] e− j n x [n] = 12π ∫−π

+πX (e j ω̂) e+ j ω̂n

= {e+ j ω̂ N +⋯+ e− j ω̂ N}

X (e j ω̂) = ∑n=−N

+N

e− j ω̂n x [n]

= e+ j ω̂N {1 +⋯+ e− j ω̂ 2N }

= e+ j ω̂N 1− e− j ω̂ (2N+1)

1 − e− j ω̂

= e+ j ω̂N e− j ω̂(2 N+1)/2

e− j ω̂ /2e+ j ω̂(2N+1)/2 − e− j ω̂(2 N+1)/2

e+ j ω̂ /2 − e− j ω̂ /2

= e+ j ω̂(2N+1)/2 − e− j ω̂(2N+1)/2

e+ j ω̂ /2 − e− j ω̂ /2 = sin(ω̂(2N+1)/2)sin(ω̂/2)

X (e j ω̂) = sin(ω̂ L/2)sin(ω̂/2)

L = 2 N+1

DL(ej ω̂) = sin(ω̂L /2)

Lsin(ω̂/2)

Dirichlet Function

= L DL(ej ω̂)

= L⋅diric(ω̂ , L)

7 Young Won Lim4/20/13DT Rect (4B)

D9(ej ω̂) = sin(ω̂9/2)

9sin (ω̂/2)

D11(ej ω̂) = sin (ω̂11/2)

11sin(ω̂/2)

D13(ej ω̂) = sin (ω̂13/2)

13sin(ω̂/2)

D9(ej ω̂)

8 zero crossings

8 zero crossings

10 zero crossings

12 zero crossings

Dirichlet Functions

2π

D10(ej ω̂) = sin(ω̂10 /2)

10sin (ω̂ /2)

D12(ej ω̂) = sin (ω̂12/2)

12sin(ω̂/2)

D14(ejω̂) = sin(ω̂14 /2)

14sin (ω̂ /2)

D10(ej ω̂)

9 zero crossings

9 zero crossings

11 zero crossings

13 zero crossings

2π

8 Young Won Lim4/20/13DT Rect (4B)

Magnitude Response

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-20 -15 -10 -5 0 5 10 15 20

9 Young Won Lim4/20/13DT Rect (4B)

Phase Response

0

0.5

1

1.5

2

2.5

3

3.5

-20 -15 -10 -5 0 5 10 15 20

10 Young Won Lim4/20/13DT Rect (4B)

11 Young Won Lim4/20/13DT Rect (4B)

RectN[n] * δN0[n] DTFS (1)

Dirichlet Function

Discrete Time Fourier Series DTFS

X [k ] = 1N ∑

n= 0

N − 1

x [n] e− j (2π /N )k n x [n] = ∑k = 0

N − 1

X [k ] e+ j(2π/N )k n

X [k ] = 1N 0

∑n=0

N 0−1

x [n]e− j (2π/N 0)k n

= 1N 0

∑n=−N

+N

x [n ]e− j (2π/N 0)k n

N 0 X [k ] = e+ j (2π N /N 0)k +⋯+ e− j (2π N /N 0)k

= e+ j (m)N k⋅e− j (m)(2N+1)k /2

e− j (m) k /2 ⋅e+ j (m)(2 N+1)k /2 − e− j (m)(2 N+1)k /2

e+ j (m)k /2 − e− j (m) k /2

= e+ j (m)N k⋅1 − e− j (m)(2 N+1) k

1− e− j (m)km = (2π/N 0)

= sin((m)(2 N+1)k /2)sin((m)k /2)

X [k ] = 1N 0

sin ((2π/N 0)(2 N+1)k /2)sin ((2π/N 0)k /2)

drcl (t , L) = sin(π Lt )Lsin(π t)

= e+ j (2π/N 0) N k⋅ 1− e− j (2π/N 0)(2N+1)k

1 − e− j (2π/N 0)k

0

1

+N−N

L = 2 N+1 N 0

12 Young Won Lim4/20/13DT Rect (4B)

RectN[n] * δN0[n] DTFS (2)

0

1

+N−N

L = 2 N+1

DL(ej ω̂) = sin(ω̂L /2)

Lsin(ω̂/2)

Dirichlet Function

Discrete Time Fourier Series DTFS

X [k ] = 1N ∑

n= 0

N − 1

x [n] e− j (2π /N )k n x [n] = ∑k = 0

N − 1

X [k ] e+ j(2π/N )k n

N 0

drcl (t , L) = sin(π Lt )Lsin(π t)

drcl (k /N 0 , (2 N+1)) =sin (π k (2 N+1)/N 0)(2 N+1)sin (π k /N 0)

X [k ] = (2 N+1)N 0

⋅drcl (k /N 0 , (2 N+1))

X [k ] = 1N 0

sin((2π/N 0)(2 N+1)k /2)sin ((2π/N 0)k /2)

= 1N 0

sin (π k (2N+1)/N 0)sin (π k /N 0)

X [k ] = LN 0

⋅drcl (k /N 0 , L)X [k ] = 1N 0

sin(π k L /N 0)sin(π k /N 0)

13 Young Won Lim4/20/13DT Rect (4B)

RectN[n] * δN0[n] DTFS (3)

0

1

+N−N

L = 2 N+1

DL(ej ω̂) = sin(ω̂L /2)

Lsin(ω̂/2)

Dirichlet Function

Discrete Time Fourier Series DTFS

X [k ] = 1N ∑

n= 0

N − 1

x [n] e− j (2π /N )k n x [n] = ∑k = 0

N − 1

X [k ] e+ j(2π/N )k n

N 0

drcl (t , L) = sin(π Lt )Lsin(π t)

X [k ] = LN 0

⋅drcl (k /N 0 , L)

X [k ] = 1N 0

sin(π k L /N 0)sin(π k /N 0)

Period : N0 (odd L), 2N0 (even L)

(L-1) zero crossings

14 Young Won Lim4/20/13DT Rect (4B)

RectN[n] * δN0[n] DTFS (4)

odd L=9 even L=10

t = 0 t =+1 t =+2t =−1t =−2 t = 0 t =+1 t =+2t =−1t =−2

Dirichlet Function

drcl (t , L) =sin(π L t)L sin(π t)

X [k ] = 916 ⋅drcl (k /16 , 9)

k=0 k=+16 k=+32k=−16k=−32 k=0 k=+16 k=+32k=−16k=−32

8 zero crossings

9 zero crossings

LL

⋯ −3, −2, −1, 0, +1, +2, +3,⋯

(L-1) zero crossings (L-1) zero crossings

15 Young Won Lim4/20/13DT Rect (4B)

RectN[n] * δN0[n] DTFS (5)

X [k ] = 916 ⋅drcl (k /16 , 9)

Period : N0 (odd L), 2N0 (even L)

k=0 k=+16 k=+32k=−16k=−32

(L-1) zero crossings

16 Young Won Lim4/20/13DT Rect (4B)

Rect2[n] * δ8[n] DTFS Example

0

1

+N−N

DL(ej ω̂) = sin(ω̂L /2)

Lsin(ω̂/2)

Dirichlet Function

Discrete Time Fourier Series DTFS

X [k ] = 1N ∑

n= 0

N − 1

x [n] e− j (2π /N )k n x [n] = ∑k = 0

N − 1

X [k ] e+ j(2π/N )k n

N 0=8 drcl (t , L) = sin(π Lt )Lsin(π t)

X [k ] = 1N 0

sin (π k (2 N+1)/N 0)sin (π k /N 0)

X [k ] = 58⋅drcl (k /8 , 5)

L = 2N+1

X [k ] = LN 0

⋅drcl (k /N 0 , L)

L = 5 (N = 2)N 0=8

X [k ] = 18sin(π k 5 /8)sin (π k /8)

Period : N0 = 8 (odd L = 5)(L – 1) = 4 zero crossings

17 Young Won Lim4/20/13DT Rect (4B)

Magnitude Response

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-20 -15 -10 -5 0 5 10 15 20

716drcl( k16 ,7)

= 116sin(π k 7/16)7sin(π k /16)

18 Young Won Lim4/20/13DT Rect (4B)

Phase Response

0

0.5

1

1.5

2

2.5

3

3.5

-20 -15 -10 -5 0 5 10 15 20

716drcl( k16 ,7)

= 116sin(π k 7/16)7sin(π k /16)

19 Young Won Lim4/20/13DT Rect (4B)

RectN[n] * δN0[n] DFT

0

1

+N−N

L = 2 N+1

DL(ej ω̂) = sin(ω̂L /2)

Lsin(ω̂/2)

Dirichlet Function

N 0 drcl (t , L) = sin(π Lt )Lsin(π t)

drcl (k /N 0 , (2 N+1)) =sin (π k /N 0(2N+1))(2 N+1)sin (π k /N 0)

X [k ] = (2 N+1)⋅drcl (k /N 0 , (2N+1))

X [k ] =sin((2π/N 0)(2N+1)k /2)

sin ((2π/N 0)k /2)

=sin (π k /N 0(2 N+1))

sin (π k /N 0)

= L⋅drcl (k /N 0 , L)=sin (π k /N 0 L)sin (π k /N 0)

Discrete Fourier Transform

X [k ] = ∑n= 0

N − 1

x [n] e− j2 /N k n x [n ] = 1N ∑

k = 0

N − 1

X [k ] e j 2/N k n

20 Young Won Lim4/20/13DT Rect (4B)

RectN[n] * δN0[n] DTFS & DFT

Discrete Time Fourier Series DTFS

X [k ] = 1N ∑

n= 0

N − 1

x [n] e− j (2π /N )k n x [n] = ∑k = 0

N − 1

X [k ] e+ j(2π/N )k n

X [k ] = LN 0

⋅drcl (k /N 0 , L)X [k ] = 1N 0

sin (π k L /N 0)sin (π k /N 0)

Discrete Fourier Transform

X [k ] = ∑n= 0

N − 1

x [n] e− j2 /N k n x [n ] = 1N ∑

k = 0

N − 1

X [k ] e j 2/N k n

X [k ] = L⋅drcl (k /N 0 , L)X [k ] =sin (π k /N 0 L)sin (π k /N 0)

Young Won Lim4/20/13

References

[1] http://en.wikipedia.org/[2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003[3] G. Beale, http://teal.gmu.edu/~gbeale/ece_220/fourier_series_02.html[4] C. Langton, http://www.complextoreal.com/chapters/fft1.pdf