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13: Resampling Filters


• Resampling

• Halfband Filters

• Dyadic 1:8 Upsampler

• Rational Resampling

• Arbitrary Resampling +

• Polynomial Approximation

• Farrow Filter +

• Summary

• MATLAB routines

DSP and Digital Filters (2017-10126) Resampling: 13 – 1 / 10

Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


Suppose we want to change the sample rate while preserving information:e.g. Audio 44.1 kHz↔48 kHz↔96 kHz

Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



Downsample:LPF to new Nyquist bandwidth: ω0 = π

K

Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines




K

Upsample:LPF to old Nyquist bandwidth: ω0 = π

K

Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines




K


K

Rational ratio: fs ×PQ

LPF to lower of old and new Nyquistbandwidths: ω0 = π

max(P,Q)

Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines




K


K



max(P,Q)

• Polyphase decomposition reduces computation by K = max(P,Q).

Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines




K


K



max(P,Q)


• The transition band centre should be at the Nyquist frequency, ω0 = πK

Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines




K


K



max(P,Q)



• Filter order M ≈ d3.5∆ω

where d is stopband attenuation in dB and ∆ω

is the transition bandwidth (Remez-exchange estimate).

Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines




K


K



max(P,Q)






• Fractional semi-Transition bandwidth, α = ∆ω2ω0

, is typically fixed.

Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines




K


K



max(P,Q)






• Fractional semi-Transition bandwidth, α = ∆ω2ω0

, is typically fixed.

e.g. α = 0.05 ⇒ M ≈ dK7πα = 0.9dK (where ω0 = π

K)

Halfband Filters


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


If K = 2 then the new Nyquist frequency isω0 = π

2 .

0 1 2 30

0.5

1

ω (rad/s)

Halfband Filters


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



2 .

We multiply ideal response sinω0nπn

by a Kaiserwindow. 0 1 2 3

0

0.5

1

ω (rad/s)

M=20β=2.5

Halfband Filters


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



2 .


by a Kaiserwindow. All even numbered points are zeroexcept h[0] = 0.5.

0 1 2 30

0.5

1

ω (rad/s)

M=20β=2.5

Halfband Filters


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



2 .



If 4 | M and we make the filter causal (×z−M2 ),

H(z) = 0.5z−M2 + z−1

∑

M2−1

r=0 h1[r]z−2r

where h1[r] = h[2r + 1− M2 ]

0 1 2 30

0.5

1

ω (rad/s)

M=20β=2.5

Halfband Filters


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



2 .




H(z) = 0.5z−M2 + z−1

∑

M2−1

r=0 h1[r]z−2r

where h1[r] = h[2r + 1− M2 ]

Half-band upsampler:

0 1 2 30

0.5

1

ω (rad/s)

M=20β=2.5

Halfband Filters


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



2 .




H(z) = 0.5z−M2 + z−1

∑

M2−1

r=0 h1[r]z−2r

where h1[r] = h[2r + 1− M2 ]


We interchange the filters with the 1:2 blockand use the commutator notation.

0 1 2 30

0.5

1

ω (rad/s)

M=20β=2.5

Halfband Filters


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



2 .




H(z) = 0.5z−M2 + z−1

∑

M2−1

r=0 h1[r]z−2r

where h1[r] = h[2r + 1− M2 ]



H1(z) is symmetrical with M2 coefficients

so we need M4 multipliers in total (input gain

of 0.5 can usually be absorbed elsewhere).

0 1 2 30

0.5

1

ω (rad/s)

M=20β=2.5

Halfband Filters


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



2 .




H(z) = 0.5z−M2 + z−1

∑

M2−1

r=0 h1[r]z−2r

where h1[r] = h[2r + 1− M2 ]



H1(z) is symmetrical with M2 coefficients

so we need M4 multipliers in total (input gain

of 0.5 can usually be absorbed elsewhere).

Computation: M4 multiplies per input sample

0 1 2 30

0.5

1

ω (rad/s)

M=20β=2.5

Dyadic 1:8 Upsampler


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


Suppose X(z): BW = 0.8π ⇔ α = 0.2 0 1 2 30

0.5

1 0.8π

ω



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


Suppose X(z): BW = 0.8π ⇔ α = 0.2

Upsample 1:2 → U(z):

0 1 2 30

0.5

1 0.8π

ω



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


Suppose X(z): BW = 0.8π ⇔ α = 0.2

Upsample 1:2 → U(z):Filter HP (z) must remove image: ∆ω = 0.2π

0 1 2 30

0.5

1 0.8π

ω

0 1 2 30

0.5

1 0.4π 0.6π

P=28

ω



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


Suppose X(z): BW = 0.8π ⇔ α = 0.2

Upsample 1:2 → U(z):Filter HP (z) must remove image: ∆ω = 0.2πFor attenuation = 60 dB, P ≈ 60

3.5∆ω= 27.3

0 1 2 30

0.5

1 0.8π

ω

0 1 2 30

0.5

1 0.4π 0.6π

P=28

ω



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


Suppose X(z): BW = 0.8π ⇔ α = 0.2


3.5∆ω= 27.3

Round up to a multiple of 4: P = 28

0 1 2 30

0.5

1 0.8π

ω

0 1 2 30

0.5

1 0.4π 0.6π

P=28

ω



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


Suppose X(z): BW = 0.8π ⇔ α = 0.2


3.5∆ω= 27.3


Upsample 1:2 → V (z): ∆ω = 0.6π⇒ Q = 12

0 1 2 30

0.5

1 0.8π

ω

0 1 2 30

0.5

1 0.4π 0.6π

P=28

ω

0 1 2 30

0.5

1 0.2π 0.8π

Q=12

ω



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


Suppose X(z): BW = 0.8π ⇔ α = 0.2


3.5∆ω= 27.3


Upsample 1:2 → V (z): ∆ω = 0.6π⇒ Q = 12

Upsample 1:2 → Y (z): ∆ω = 0.8π⇒ R = 8

0 1 2 30

0.5

1 0.8π

ω

0 1 2 30

0.5

1 0.4π 0.6π

P=28

ω

0 1 2 30

0.5

1 0.2π 0.8π

Q=12

ω

0 1 2 30

0.5

1 0.1π 0.9π

R=8

ω



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


Suppose X(z): BW = 0.8π ⇔ α = 0.2


3.5∆ω= 27.3


Upsample 1:2 → V (z): ∆ω = 0.6π⇒ Q = 12

Upsample 1:2 → Y (z): ∆ω = 0.8π⇒ R = 8[diminishing returns + higher sample rate]

0 1 2 30

0.5

1 0.8π

ω

0 1 2 30

0.5

1 0.4π 0.6π

P=28

ω

0 1 2 30

0.5

1 0.2π 0.8π

Q=12

ω

0 1 2 30

0.5

1 0.1π 0.9π

R=8

ω



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


Suppose X(z): BW = 0.8π ⇔ α = 0.2


3.5∆ω= 27.3


Upsample 1:2 → V (z): ∆ω = 0.6π⇒ Q = 12


Multiplication Count:(

1 + P4

)

× fx +Q

4 × 2fx +R4 × 4fx = 22fx

0 1 2 30

0.5

1 0.8π

ω

0 1 2 30

0.5

1 0.4π 0.6π

P=28

ω

0 1 2 30

0.5

1 0.2π 0.8π

Q=12

ω

0 1 2 30

0.5

1 0.1π 0.9π

R=8

ω



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


Suppose X(z): BW = 0.8π ⇔ α = 0.2


3.5∆ω= 27.3


Upsample 1:2 → V (z): ∆ω = 0.6π⇒ Q = 12


Multiplication Count:(

1 + P4

)

× fx +Q

4 × 2fx +R4 × 4fx = 22fx

0 1 2 30

0.5

1 0.8π

ω

0 1 2 30

0.5

1 0.4π 0.6π

P=28

ω

0 1 2 30

0.5

1 0.2π 0.8π

Q=12

ω

0 1 2 30

0.5

1 0.1π 0.9π

R=8

ω

0 1 2 30

0.5

1 0.15π

M=110

ω

Alternative approach using direct 1:8 upsampling:∆ω = 0.05π ⇒ M = 110⇒ 111fx multiplications (using polyphase)

Rational Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


To resample by PQ

do 1:Pthen LPF, then Q:1.

Rational Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


To resample by PQ


Resample by PQ

⇒ ω0 = πmax(P,Q)

Rational Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


To resample by PQ


Resample by PQ


∆ω , 2αω0 = 2απmax(P,Q)

Rational Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


To resample by PQ


Resample by PQ



Polyphase: H(z) =∑P−1

p=0 z−pHp(zP )

Rational Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


To resample by PQ


Resample by PQ




p=0 z−pHp(zP )

Commutate coefficients:v[s] uses Hp(z) with p = smodP

Rational Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


To resample by PQ


Resample by PQ




p=0 z−pHp(zP )


Keep only every Qth output:

Rational Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


To resample by PQ


Resample by PQ




p=0 z−pHp(zP )


Keep only every Qth output:y[i] uses Hp(z) with p = QimodP

Rational Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


To resample by PQ


Resample by PQ




p=0 z−pHp(zP )



Multiplication Count:H(z): M + 1 ≈ 60 [dB]

3.5∆ω= 2.7max(P,Q)

α

Rational Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


To resample by PQ


Resample by PQ




p=0 z−pHp(zP )




3.5∆ω= 2.7max(P,Q)

α

M + 1 coeficients in all

Rational Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


To resample by PQ


Resample by PQ




p=0 z−pHp(zP )




3.5∆ω= 2.7max(P,Q)

α

Hp(z): R+ 1 = M+1P

= 2.7α

max(

1, Q

P

)


Rational Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


To resample by PQ


Resample by PQ




p=0 z−pHp(zP )




3.5∆ω= 2.7max(P,Q)

α

Hp(z): R+ 1 = M+1P

= 2.7α

max(

1, Q

P

)


Multiplication rate: 2.7α

max(

1, QP

)

× fy

Rational Resampling


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


To resample by PQ


Resample by PQ




p=0 z−pHp(zP )




3.5∆ω= 2.7max(P,Q)

α

Hp(z): R+ 1 = M+1P

= 2.7α

max(

1, Q

P

)


Multiplication rate: 2.7α

max(

1, QP

)

× fy = 2.7α

max (fy, fx)

Arbitrary Resampling +


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


Sometimes need very large P and Q:e.g. 44.1 kHz

48 kHz = 147160



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



48 kHz = 147160

Multiplication rate OK:2.7max(fy, fx)

α



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



48 kHz = 147160


α

However # coefficients: 2.7max(P,Q)α



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



48 kHz = 147160


α


Alternatively, use any large integer Pand round down to the nearest sample:

E.g. for y[i] at time iQP

use hp[r]where p = (⌊iQ⌋)mod P



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



48 kHz = 147160


α





Equivalent to converting to analog withzero-order hold and resampling at fy = P

Q.



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



48 kHz = 147160


α






Q.

Zero-order hold convolves with rectangular 1P

-wide window ⇒ multiplies

periodic spectrum bysin Ω

2PΩ

2P

.



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



48 kHz = 147160


α






Q.




2PΩ

2P

. Resampling aliases Ω to Ωmod 2PπQ

.



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



48 kHz = 147160


α






Q.




2PΩ

2P


.

Unit power component at Ω1 gives alias components with total power:

sin2 Ω1

2P

∑

∞

n=1

(

2P2nPπ+Ω1

)2

+(

2P2nPπ−Ω1

)2

≈ω2

1

4P 2

2π2

6π2 =Ω2

1

12P 2



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



48 kHz = 147160


α






Q.




2PΩ

2P


.

Unit power component at Ω1 gives alias components with total power:

sin2 Ω1

2P

∑

∞

n=1

(

2P2nPπ+Ω1

)2

+(

2P2nPπ−Ω1

)2

≈ω2

1

4P 2

2π2

6π2 =Ω2

1

12P 2

For worst case, Ω1 = π, need P = 906 to get −60 dB /

Polynomial Approximation


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


Suppose P = 50 and H(z) has order M = 249H(z) is lowpass filter with ω0 ≈ π

50

0 0 0 0 0



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



50

0 1 2 3-80-60-40-20

0

ω

0 50 100 150 2000

0.5

1M=249, β=6.5

m

0 0 0 0 0



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



50

Split into 50 filters of length R + 1 = M+1P

= 5:

0 1 2 3-80-60-40-20

0

ω

0 50 100 150 2000

0.5

1M=249, β=6.5

m

0 0 0 0 0



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



50


= 5:hp[0] is the first P samples of h[m]

0 1 2 3-80-60-40-20

0

ω

0 50 100 150 2000

0.5

1

hp[0]

M=249, β=6.5

m

0 0 0 0 0



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



50


= 5:hp[0] is the first P samples of h[m]hp[1] is the next P samples, etc.

0 1 2 3-80-60-40-20

0

ω

0 50 100 150 2000

0.5

1

hp[0] h

p[1] h

p[2] h

p[3] h

p[4]

M=249, β=6.5

m

0 0 0 0 0



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



50


= 5:hp[0] is the first P samples of h[m]hp[1] is the next P samples, etc.hp[r] = h[p+ rP ]

0 1 2 3-80-60-40-20

0

ω

0 50 100 150 2000

0.5

1

hp[0] h

p[1] h

p[2] h

p[3] h

p[4]

M=249, β=6.5

m

0 0 0 0 0



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



50



Use a polynomial of order L toapproximate each segment:hp[r] ≈ fr(

pP) with 0 ≤ p

P< 1

0 1 2 3-80-60-40-20

0

ω

0 50 100 150 2000

0.5

1

hp[0] h

p[1] h

p[2] h

p[3] h

p[4]

M=249, β=6.5

m

0 0 0 0 0



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



50




pP) with 0 ≤ p

P< 1

h[m] is smooth, so errors are low.E.g. error < 10−3 for L = 4

0 1 2 3-80-60-40-20

0

ω

0 50 100 150 2000

0.5

1

hp[0] h

p[1] h

p[2] h

p[3] h

p[4]

M=249, β=6.5

m

0 50 100 150 200

-1

0

1x 10

-3

L=4

erro

r



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



50




pP) with 0 ≤ p

P< 1


• Resultant filter almost as good

0 1 2 3-80-60-40-20

0

ω

0 50 100 150 2000

0.5

1

hp[0] h

p[1] h

p[2] h

p[3] h

p[4]

M=249, β=6.5

m

0 50 100 150 200

-1

0

1x 10

-3

L=4

erro

r0 1 2 3

-80-60-40-20

0 Farrow with L=4

ω

ω0



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



50




pP) with 0 ≤ p

P< 1


• Resultant filter almost as good• Instead of M + 1 = 250

coefficients we only need(R+ 1)(L+ 1) = 25

0 1 2 3-80-60-40-20

0

ω

0 50 100 150 2000

0.5

1

hp[0] h

p[1] h

p[2] h

p[3] h

p[4]

M=249, β=6.5

m

0 50 100 150 200

-1

0

1x 10

-3

L=4

erro

r0 1 2 3

-80-60-40-20

0 Farrow with L=4

ω

ω0



• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



50




pP) with 0 ≤ p

P< 1


• Resultant filter almost as good• Instead of M + 1 = 250

coefficients we only need(R+ 1)(L+ 1) = 25

whereR+ 1 = 2.7

αmax

(

1, Q

P

)

0 1 2 3-80-60-40-20

0

ω

0 50 100 150 2000

0.5

1

hp[0] h

p[1] h

p[2] h

p[3] h

p[4]

M=249, β=6.5

m

0 50 100 150 200

-1

0

1x 10

-3

L=4

erro

r0 1 2 3

-80-60-40-20

0 Farrow with L=4

ω

ω0

Farrow Filter +


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


Filter coefficients depend on fractional part of iQP

:

∆[i] = iQP− n where n =

⌊

iQP

⌋

R+ 1 =M+1

P= 5

Farrow Filter +


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



:


⌊

iQP

⌋

y[i] =∑R

r=0 fr(∆[i])x[n− r]

R+ 1 =M+1

P= 5

Farrow Filter +


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



:


⌊

iQP

⌋

y[i] =∑R

r=0 fr(∆[i])x[n− r]

where fr(∆) =∑L

l=0 bl[r]∆l

R+ 1 =M+1

P= 5

Farrow Filter +


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



:


⌊

iQP

⌋

y[i] =∑R

r=0 fr(∆[i])x[n− r]

where fr(∆) =∑L

l=0 bl[r]∆l

y[i] =∑R

r=0

∑L

l=0 bl[r]∆[i]lx[n− r]

R+ 1 =M+1

P= 5

Farrow Filter +


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



:


⌊

iQP

⌋

y[i] =∑R

r=0 fr(∆[i])x[n− r]

where fr(∆) =∑L

l=0 bl[r]∆l

y[i] =∑R

r=0

∑L


=∑L

l=0 ∆[i]l∑R

r=0 bl[r]x[n− r]

R+ 1 =M+1

P= 5

Farrow Filter +


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



:


⌊

iQP

⌋

y[i] =∑R

r=0 fr(∆[i])x[n− r]

where fr(∆) =∑L

l=0 bl[r]∆l

y[i] =∑R

r=0

∑L


=∑L

l=0 ∆[i]l∑R

r=0 bl[r]x[n− r]

=∑L

l=0 ∆[i]lvl[n]

R+ 1 =M+1

P= 5

Farrow Filter +


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



:


⌊

iQP

⌋

y[i] =∑R

r=0 fr(∆[i])x[n− r]

where fr(∆) =∑L

l=0 bl[r]∆l

y[i] =∑R

r=0

∑L


=∑L

l=0 ∆[i]l∑R

r=0 bl[r]x[n− r]

=∑L

l=0 ∆[i]lvl[n]where vl[n] = bl[n] ∗ x[n]

R+ 1 =M+1

P= 5

Farrow Filter +


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines



:


⌊

iQP

⌋

y[i] =∑R

r=0 fr(∆[i])x[n− r]

where fr(∆) =∑L

l=0 bl[r]∆l

y[i] =∑R

r=0

∑L


=∑L

l=0 ∆[i]l∑R

r=0 bl[r]x[n− r]

=∑L


Horner’s Rule:y[i] = v0[n]+∆ (v1[n] + ∆ (v2[n] + ∆ (· · · )))

R+ 1 =M+1

P= 5

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:


⌊

iQP

⌋

y[i] =∑R

r=0 fr(∆[i])x[n− r]

where fr(∆) =∑L

l=0 bl[r]∆l

y[i] =∑R

r=0

∑L


=∑L

l=0 ∆[i]l∑R

r=0 bl[r]x[n− r]

=∑L



Multiplication Rate:Each Bl(z) needs R+ 1 per input sampleHorner needs L per output sample

R+ 1 =M+1

P= 5

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:


⌊

iQP

⌋

y[i] =∑R

r=0 fr(∆[i])x[n− r]

where fr(∆) =∑L

l=0 bl[r]∆l

y[i] =∑R

r=0

∑L


=∑L

l=0 ∆[i]l∑R

r=0 bl[r]x[n− r]

=∑L




R+ 1 =M+1

P= 5

Total: (L+ 1) (R + 1) fx + Lfy = 2.7(L+1)α

max(

1, fxfy

)

fx + Lfy

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:


⌊

iQP

⌋

y[i] =∑R

r=0 fr(∆[i])x[n− r]

where fr(∆) =∑L

l=0 bl[r]∆l

y[i] =∑R

r=0

∑L


=∑L

l=0 ∆[i]l∑R

r=0 bl[r]x[n− r]

=∑L

l=0 ∆[i]lvl[n]where vl[n] = bl[n] ∗ x[n][like a Taylor series expansion]



R+ 1 =M+1

P= 5

R+1 ≈2.7

αmax

(

1,Q

P

)

Total: (L+ 1) (R + 1) fx + Lfy = 2.7(L+1)α

max(

1, fxfy

)

fx + Lfy

Summary


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• MATLAB routines


• Transition band centre at ω0

ω0 = the lower of the old and new Nyquist frequencies Transition width = ∆ω = 2αω0, typically α ≈ 0.1

Summary


• Resampling






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• Summary

• MATLAB routines




• Factorizing resampling ratio can reduce computation halfband filters very efficient (half the coefficients are zero)

Summary


• Resampling






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• Summary

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• Rational resampling ×PQ

# multiplies per second: 2.7α

max (fy, fx) # coefficients: 2.7

αmax (P, Q)

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αmax (P, Q)

• Farrow Filter approximate filter impulse response with polynomial segments arbitrary, time-varying, resampling ratios

# multiplies per second: 2.7(L+1)α

max (fy, fx)×fxfy

+ Lfy

⊲ ≈ (L+ 1) fxfy

times rational resampling case

# coefficients: 2.7α

max (P, Q)× L+1P

coefficients are independent of fy when upsampling

Summary


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines








αmax (P, Q)

• Farrow Filter approximate filter impulse response with polynomial segments arbitrary, time-varying, resampling ratios

# multiplies per second: 2.7(L+1)α

max (fy, fx)×fxfy

+ Lfy

⊲ ≈ (L+ 1) fxfy

times rational resampling case

# coefficients: 2.7α

max (P, Q)× L+1P

coefficients are independent of fy when upsampling

For further details see Mitra: 13 and Harris: 7, 8.

MATLAB routines


• Resampling






• Farrow Filter +

• Summary

• MATLAB routines


gcd(p,q) Find αp+ βq = 1 for coprime p, qpolyfit Fit a polynomial to datapolyval Evaluate a polynomialupfirdn Perform polyphase filtering

resample Perform polyphase resampling

13: resampling filters halfband filters polynomial ... · pdf file13: resampling filters 13:...

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