Young Won Lim6/5/12

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Anti-aliasing Prefilter (6B)

Young Won Lim6/5/12

Copyright (c) 2012 Young W. Lim.

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6B Prefilter 3 Young Won Lim6/5/12

Sampler

T T

τ

x̂ (t) = ∑n =−∞

+∞

x(nT ) δ(t−nT ) x̂ (t) ≈ ∑n =−∞

+∞

x (nT ) p (t−nT )

x̂(t)

X̂ ( f ) = ∫−∞

+∞

x̂ (t ) e− j 2π f t dt

Ideal Sampling Practical Sampling

6B Prefilter 4 Young Won Lim6/5/12

Zero Order Hold (ZOH)

T T

τrect( t−T /2T )

xZOH (t ) = ∑n=−∞

+∞

x [n]⋅rect( t−T /2−nTT )

1/T

6B Prefilter 5 Young Won Lim6/5/12

Square Wave CTFT

C k =1T ∫0

Tx (t ) e− j k ω0 t dt x(t) = ∑

n=0

∞

C k e+ j k ω0 t

Continuous Time Fourier Series

C k =1

T 0∫−T 0 /2

+T 0 /2xT 0(t ) e

− j k ω0 t dt

C k T 0 = ∫−T 0/2+T 0/2 xT 0(t) e

− j k ω0 t dt

= ∫−T 0/ 2+T 0 /2 e− j k ω0 t dt = [ −1j k ω0 e− j k ω0 t]−T /2

+T / 2

= −e− j k ω0T / 2−e+ j k ω0T /2

j k ω0=sin (k ω0T /2)

k ω0/2

−T 02 +

T 02

2ω0−2ω0

∣ak∣

ω0−ω0

1T 0

2ω0−2ω0

∣ak T 0∣1

ω0−ω0

ω0 =2πT 0

Fundamental Frequency

k ω0

sin (k ω0T /2)k ω0/2

T 0

6B Prefilter 6 Young Won Lim6/5/12

CTFT and CTFS

X j = ∫−∞∞

x t e− j t d t x t =12 ∫−∞

∞X j e j t d

C k =1T ∫0

Tx (t ) e− j k ω0 t dt x(t) = ∑

n=0

∞

C k e+ j k ω0 t

Continuous Time Fourier Series

Continuous Time Fourier Transform Aperiodic Continuous Time Signal

Periodic Continuous Time Signal

−T2 +T2−

T 02 +

T 02

−T2 +T2

6B Prefilter 7 Young Won Lim6/5/12

CTFT ← CTFSAperiodic Continuous Time Signal

Periodic Continuous Time Signal

−T2 +T2

−T2+T2

−T 02 +

T 02

x(t )

xT 0(t) = ∑n=−∞

+∞

x(t − nT 0)

xT 0(t) → x(t )

ω0 =2πT 0

→ 0

As T 0 → ∞ ,

Continuous Time Fourier Series

Continuous Time Fourier Transform

T 0

T 0 → ∞

6B Prefilter 8 Young Won Lim6/5/12

CTFT and CTFS as

−T2 +

T2 +

T 02

+T 0−T 02

−T 0 +2T0−2T0

−T2 +

T2 +

T 02

+T 0−T 02

−T 0

−T2 +

T2 +

T 02

−T 02

T 0 = 2T

T 0 = 4T

T 0 = 8T

2ω0−2ω0

4ω0−4ω0

8ω0−8ω0

∣ak T 0∣

∣ak T 0∣

∣ak T 0∣

T 0 → ∞ ,

6B Prefilter 9 Young Won Lim6/5/12

Sampling (1)

T T

τ

T

τ

T T

τ

X

=

X=

Ideal Sampling Practical Sampling

t

t

Tt

6B Prefilter 10 Young Won Lim6/5/12

Sampling (2)

T

T

T

Ideal Sampling Frequency Domain

X

=

*=

1T

ω

t

tω

t

ω

A

AT

2ω0−2ω0 ω0−ω0

2ω0−2ω0 ω0−ω0

ω0 =2πT

6B Prefilter 11 Young Won Lim6/5/12

Practical Sampling Frequency Domain

Sampling (3)

T

τ

T

τ

T

τ

2ω0−2ω0

∣ak∣

ω0−ω0 k ω0

1Tsin (k ω0T /2)

k ω0/2

X

=

*=

2ω0−2ω0 ω0−ω0

t

t

t

A

AT

ω0 =2πT

6B Prefilter 12 Young Won Lim6/5/12

CTFT of Sampled Signal

x̂(t) = ∑n =−∞

+∞

x (nT ) δ( t−nT ) X̂ ( f ) = ∫−∞

+∞

x̂( t) e− j 2π f t dt

= ∫−∞

+∞

∑n =−∞

+∞

x(nT ) δ(t−nT ) e− j 2π f t dt

= ∑n =−∞

+∞

x(nT ) e− j 2π f T n dt

X̂ ( f ) = ∑n =−∞

+∞

x(nT ) e− j 2π f T n

Tt

ω0−

2πT

−2 2πT

+2πT

+2⋅2πT

AT

CTFT

CTFT

x̂(t) = ∑n =−∞

+∞

x (nT ) δ( t−nT )

6B Prefilter 13 Young Won Lim6/5/12

f s =1T 2π f s =

2πT = ω0

Periodicity in Frequency

Tt

ω0−

2πT

−2 2πT

+2πT

+2⋅2πT

AT

e− j2π( f + f s)T ne− j2π( f + f s)T n = e− j 2π( f )T n f sT = 1

X̂ ( f ) = X̂ ( f + f s)

Period = Sampling Frequency f s

X̂ ( f ) = ∑n =−∞

+∞

x(nT ) e− j 2π f T nCTFT

x̂(t) = ∑n =−∞

+∞

x (nT ) δ(t−nT )

f s

2π f = ω

6B Prefilter 14 Young Won Lim6/5/12

Time Sequence

Convert to Time Sequencex (t)

x̂ (t)x [n]

p(t ) = ∑n =−∞

+∞

δ(t − nT ) T Sampling Period

Ideal Sampling

Ttt

Tt

T

Tt

x (t) x̂ (t)

p(t ) x [n ] Time Sequence

6B Prefilter 15 Young Won Lim6/5/12

DTFT of a Time Sequence

X̂ ( f ) = ∑n =−∞

+∞

x(nT ) e− j 2π f T n

tω

−2πT

−2 2πT

+2πT

+2⋅2πT

AT

CTFTx̂(t) = ∑

n =−∞

+∞

x (nT ) δ( t−nT )

tω

−2πT

−2 2πT

+2πT

+2⋅2πT

AT

X̂ ( f ) = ∑n =−∞

+∞

x [n] e− j 2π f T nDTFT

x [n]

x̂( t)

x [n] = x (nT )

6B Prefilter 16 Young Won Lim6/5/12

Discrete Time Fourier Transform (1)

2π f T = 2π f1/T = 2πff s

= ω̂

Normalized Angular Frequency

X̂ (e j ω̂) = ∑n =−∞

+∞

x [n] e− j ω̂ n

tω

−2πT

−2 2πT

+2πT

+2⋅2πT

AT

X̂ ( f ) = ∑n =−∞

+∞

x [n] e− j 2π f T nDTFT

x [n]

x [n] = x (nT )

ω̂ = ω⋅T−2π−4π +2π +4π

AT

DTFTx [n]

6B Prefilter 17 Young Won Lim6/5/12

f s =1T 2π f s =

2πT = ω0

Discrete Time Fourier Transform (2)

X̂ ( f ) = ∑n =−∞

+∞

x(nT ) e− j 2π f T n

Tt

ω0−

2πT

−2 2πT

+2πT

+2⋅2πT

AT

X̂ (e j ω̂) = ∑n =−∞

+∞

x(nT ) e− j2π ω̂ n

Normalized Angular Frequency

f0− f s−2 f s + f s +2 f s

f / f s0−1−2 +1 +2

ω̂ = 2π f / f s0−2π−4π +2π +4π

ω = 2π f0−2π f s−4π f s +2π f s +4π f s

Nyquist Interval

2π f T = 2π f1/T = 2πff s

= ω̂

6B Prefilter 18 Young Won Lim6/5/12

Fourier Series

x̂(t) = ∑n =−∞

+∞

x (nT ) δ( t−nT )

x(nT ) = 1f s∫+ f s/2

− f s/2 X̂ ( f ) e+ j 2π f T n df

= ∫−π

+π

X̂ (ω) e+ jω n d ω2π

ω = 2π f / f sd ff s

= d ω2π

Tt

ω0−

2πT

−2 2πT

+2πT

+2⋅2πT

AT

X̂ ( f ) = ∑n =−∞

+∞

x(nT ) e− j 2π f T nCTFT

CTFS

X̂ ( f )

x(nT )

Continuous Periodic Function

Fourier Series Coefficients

X̂ ( f ) = ∑n =−∞

+∞

x(nT ) e− j 2π f T n

6B Prefilter 19 Young Won Lim6/5/12

Numerical Approximation

X ( f ) = ∫−∞

+∞

x(t) e+ j 2π f t d t

≈ ∑n =−∞

+∞

x (n T ) e− j 2π f T n⋅T

X ( f ) ≈ T X̂ ( f )

X ( f ) = limT → 0

T X̂ ( f )

T

ft

Ax(t)

Tt

ω0−

2πT

−2 2πT

+2πT

+2⋅2πT

AT

x̂(t)

Tt

x̂(t) = ∑n =−∞

+∞

x (nT ) δ( t−nT ) X̂ ( f ) = ∑n =−∞

+∞

x(nT ) e− j 2π f T nCTFT

CTFT

6B Prefilter 20 Young Won Lim6/5/12

Spectrum Replication (1)

T

T

T

Ideal Sampling

X

= t

t

t

x̂(t) = ∑n =−∞

+∞

x (nT ) δ(t−nT )

s(t) = ∑n =−∞

+∞

δ(t−nT )

x( t)

x̂( t)

s( t) = 1T ∑m =−∞

+∞

e+ j 2π m f s t

x̂( t) = 1T ∑n =−∞

+∞

x (t ) e+ j 2π m f s t

Shift Property

X̂ ( f ) = 1T ∑n =−∞

+∞

X ( f −m f s)

6B Prefilter 21 Young Won Lim6/5/12

Spectrum Replication (2)

Frequency Domain

*=−

2πT

−2 2πT

+2πT

+2⋅2πT

1T

0

ω

ω0

ω

−2πT

−2 2πT

+2πT

+2⋅2πT

A

ATX̂ ( f ) = 1

T ∑n =−∞+∞

X ( f −m f s)

= ∫−∞

+∞

X ( f − f ' )S ( f ' ) d f '

S ( f ) = 1T ∑m =−∞

+∞

δ( f −m f s)

= 1T ∑m =−∞+∞

∫−∞

+∞

X ( f − f ' )δ( f '−m f s) d f '

X ( f )

X̂ ( f )

S ( f )

Convolution in Frequency

X̂ ( f ) = X ( f ) ∗ S ( f )

6B Prefilter 22 Young Won Lim6/5/12

Ideal PrefilterH(f)

Ideal Sampler- ADC

Analogsignal Bandlimitedsignal

Sampledsignal

X i ( f ) X ( f ) X̂ ( f )

f s

0 12 f s

24 f s

34 f s

0−12 f s + f s0− f s

6B Prefilter 23 Young Won Lim6/5/12

6B Prefilter 24 Young Won Lim6/5/12

Young Won Lim6/5/12

References

[1] http://en.wikipedia.org/[2] J.H. McClellan, et al., Signal Processing First, Pearson Prentice Hall, 2003[3] A “graphical interpretation” of the DFT and FFT, by Steve Mann[4] R. G. Lyons, Understanding Digital Signal Processing, 1997[5] AVR121: Enhancing ADC resolution by oversampling[6] S.J. Orfanidis, Introduction to Signal Processing

www.ece.rutgers.edu/~orfanidi/intro2sp

슬라이드 1슬라이드 2슬라이드 3슬라이드 4슬라이드 5슬라이드 6슬라이드 7슬라이드 8슬라이드 9슬라이드 10슬라이드 11슬라이드 12슬라이드 13슬라이드 14슬라이드 15슬라이드 16슬라이드 17슬라이드 18슬라이드 19슬라이드 20슬라이드 21슬라이드 22슬라이드 23슬라이드 24슬라이드 25