Polynomial-Based Interpolation Filters for DSP Applications

Tampere University of Technology

1

POLYNOMIAL-BASED INTERPOLATION FILTERS FOR DSP APPLICATIONS DESIGN, IMPLEMENTATION, AND APPLICATIONS Djordje Babic, Jussi Vesma, Tapio Saramäki, Vesa Lehtinen, Markku Renfors Tampere University of Technology, Finland

• This pile of lecture notes is based on the research work done by Dr. Tech. Jussi Vesma, Prof. Tapio Saramäki, Dr. Tech. Djordje Babic, and Prof. Markku Renfors. • The lecture notes have been prepared by Prof. Saramäki and Dr. Vesma, and updated by Dr. Babic and M.Sc. Vesa Lehtinen.


POLYNOMIAL-BASED INTERPOLATION FILTERS FOR DSP APPLICATIONS DESIGN, IMPLEMENTATION, AND APPLICATIONS Speaker: Vesa Lehtinen Tampere University of Technology, Finland e-mail: [email protected]

Contents 1. Interpolation Filters 2. Fractional-Delay Filters 3. Lagrange Interpolation 4. Analog Model for Interpolation Filter 5. Polynomial-Based Interpolation Filters 6. Design 7. Applications

2


3

INTERPOLATION FILTERS • In many DSP applications there is a need to know the value of a signal also between the existing discretetime samples x(n). • Special interpolation filters can be used to compute new sample values y(l)=ya(tl) at arbitrary points t l =(nl +µl)Tin. • Here ya(t) approximates the original continuous-time signal xa(t). • The output sample time is thus determined by the fractional interval (or delay) µl∈[0,1) and the integer index nl. • Terms "interpolator" and "fractional delay (FD) filter" are also used in the literature. ya(t)

= x(n) = y(l)

µl-1Tin nl-1Tin

tl-1

µlTin nlTin

µl+1Tin tl

nl+1Tin

Fig. 1. Interpolation in the time domain.

tl+1

Time


4

Terminology pitfalls: Mathematical interpolation vs. interpolation in DSP vs. decimation • In mathematics, interpolation stands for constructing a smooth curve that travels through all given points. o Constructing a curve that is as close as possible but does not include all the given points is referred to as approximation. Cf., e.g., leastsquares approximation. • In DSP, upsampling + antiimage filtering commonly referred to as “interpolation” (vs. decimation). o Not all DSP “interpolators” interpolate in the mathematical sense. o Filters that interpolate in the math sense are also known as Nyquist filters (recall the zero intersymbol interference property). o It is the antiimage filter that performs the interpolation, not the upsampler. • In decimation by fractional (nonintegral) ratios, output samples are generated between the input samples. => Fractional decimation incorporates interpolation / approximation for the purpose of antialiasing.


5

Applications for Interpolation Filters • Timing adjustment in all-digital receivers (symbol synchronization) • Time delay estimation • Conversion between arbitrary sample rates • Conversion between uniform and nonuniform sample grids • Echo cancellation • Phased array antenna systems • Speech coding and synthesis • Computer simulation of continuous-time systems • Image processing (image resizing and rotation)


6

Statement of the general interpolation problem for uniformly sampled input x(n)

Interpolation filter

y(l)

h(k,µl) nl

µl

Fig. 2. Simplified block diagram for interpolation filter.

• The input parameters nl and µl are used to determine the time instant tl for the l th output sample y(l)=ya(tl) as t l =(nl +µl)Tin. • Given tl, these parameters are determined by nl = tl / Tin  and µl = tl / Tin − tl / Tin 

(1)

• After knowing nl and µl, the interpolation filter calculates y(l) according to the convolution

y (l ) =

N / 2 −1

∑

k =− N / 2

x(nl − k )h(k , µl ).

(2)

where N is the length and h(k, µl) is the (discrete-time) impulse response of the interpolation filter.


7

Interpretation of the convolution sum

• First, the location of the existing sample preceding (or occurring at) the new sampling instant tl is determined and denoted by nl. • Based on the location of this sample, the existing samples located at n=nl −N/2+1, nl −N/2+2,···, nl +N/2 are used. This means that there are N/2 existing discrete-time samples before and after the desired new time instant tl. • Second, the distance between tl and nlTin is measured as a fraction of Tin, giving the fractional interval µl. • In the above convolution, µl determines the coefficient values h(k, µl). The purpose is to determine them in such a way that y(l) is found according to some criterion to be discussed next.


8

Requirements for the impulse-response coefficients h((k,, µl)

1. h(k, µl) should be optimized such that y(l)≈xa((nl +µl)Tin) for all values of µl ∈[0,1) and l. 2. The fundamental idea in the optimization is that ya(t) approximates the original continuous-time signal xa(t) according to some time-domain or frequency-domain criterion. 3. Frequency-domain criteria are preferred for DSP applications. 4. The interpolation filter should have an efficient digital implementation structure.


9

Filter Synthesis and Analysis Problem • The problem is now to design a filter which calculates the interpolated samples y(l) for a given value of µl based on the above discussion. • And to analyze the performance of the filter (what is the error between ya(t) and xa(t)). • Note that the direct analysis of the above interpolation filter is difficult, because the system is time-varying. Three approaches to find a solution

1. The frequency response of the ideal fractional delay is e−jωµ, find a FIR filter to approximate this response (FD filter approach) for each value of µ. 2. Use some classical interpolation method to calculate y(l), e.g., Lagrange or B-spline interpolation (timedomain approach). 3. Utilize the continuous-time (analog) model for the interpolation filter (frequency-domain approach).


10

FD FILTERS • There are several synthesis methods [1] to design FIR filters having a phase delay of Dint+µ, where Dint =N/2−1 is the integer delay determined by the length of the filter N and µ∈[0,1) is the fractional delay. • The amplitude response of this FIR filter should approximate unity on the given frequency band. • Good solution when µ is fixed. Otherwise, several FD FIR filters are designed and coefficients are stored in a lookup table. ⇒ Need for a large memory. • See Matlab demo #1.


11

Example: FD Filter • The length of the filter N=8, Ωp =[0, 0.75π], six different filters having µ=0, 0.1, 0.2, 0.3, 0.4, and 0.5 1 0.9

Amplitude Response

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Angular Frequency ω / π

0.8

0.9

1

Fig. 3. Amplitude response for the example FD filter. 3.5

Phase Delay Response

3.4

3.3

3.2

3.1

3 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Angular Frequency ω / π

0.8

Fig. 4. Phase delay response.

0.9

1


12

LAGRANGE INTERPOLATION • In Lagrange interpolation, the interpolated signal ya(t) is a piecewise polynomial of an odd degree M. ya((n+µ)Tin)=ŷn((n+µ)Tin) I.e., the polynomial ŷn(t) is used for t∈[nTin, (n+1)Tin). • The polynomial pieces ŷn(t) are defined such that ŷn(kTin)=x(kTin) for N=M+1 consecutive sample points k=n−N/2+1,…,n+N/2. Notice that these points extend beyond the range in which ŷn(t) is applied. • Disadvantage: Does not utilize the frequency-domain information of the input signal x(n). • Lagrange interpolation is discussed in more detail later on. • See Matlab demo #2.


13

Example: Lagrange Interpolation

• How to obtain y(l)? ZERO-ORDER HOLD

ya(t)

= x(n) = y(l)

µlTin

nl (nl+µl)Tin nl+1

LINEAR INTERPOLATION ya(t)

= x(n) = y(l)

µlTin

nl (nl+µl)Tin nl+1

CUBIC LAGRANGE INTERPOLATION = x(n) = y(l)

ya(t)

µlTin

nl (nl+µl)Tin nl+1

Fig. 5. Lagrange interpolation.


14

ANALOG MODEL FOR INTERPOLATION FILTER x(n)

xs(t)

DAC

y(l)

ya(t)

ha(t)

Sample at tl = ( n l + µ l ) Tin Fig. 6. Analog model for interpolation filter.

• Interpolation is a reconstruction problem where the approximating signal ya(t) is reconstructed based on the existing discrete-time samples x(n). • Therefore, one useful way to model discrete-time interpolation filters is to use the analog system shown in Fig. 6. • In this system, x(n) is first converted to the sequence of weighted and shifted Dirac impulses xs(t). • xs(t) is then filtered using an analog, linear timeinvariant (LTI) reconstruction filter ha(t) resulting in the following convolution: ∞

ya (t ) =

∞

∫ x (τ )h (t − τ )dτ = ∑ x(k )h (t − kT s

−∞

a

a

k = −∞

in ).

(3)


15

Why to Use the Analog Model? • Interpolation is generally considered as a time-domain polynomial fitting problem which is not a very practical approach for DSP applications. This is because the time-domain characteristics of the signal are not known usually. What is known is the frequency band of the signal. • The use of the analog model turns interpolation from a time-domain problem to a frequency-domain filtering problem. • Various interpolation methods can, indeed, be expressed as convolution when the input samples are uniformly spaced.


16

Role of ha(t) in the Frequency Domain • How to design ha(t) in the frequency domain? • Such that it preserves the signal components in the range [0,Fin /2] and attenuates them elsewhere.

X a( f )

f Fin/2 Fig. 7. The spectrum of the original continuous-time signal.

X(e j2π f /Fin) Ha( j2π f )

f Fin/2

Fin

2Fin

Fig. 8. The spectrum of the input signal x(n) and the frequency response of the reconstruction filter ha(t).


17

Farrow structure – traditional approach • The main idea of this part is the understanding of the special implementation form of polynomial-based interpolation filters. • y(l) is then obtained by sampling ya(t) at tl giving y ( l ) = y a ( tl ) =

N / 2 −1

∑ x(n

l

− k )ha (( k + µl )Tin ) .

k =−N / 2

(4)

• It is assumed that ha(t) is zero outside the interval −NTin /2≤t≤NTin /2. • By comparing (2) and (4), it can be seen that the impulse responses of the analog and discrete-time filters are related as follows: h( k , µl ) = ha((k + µl )Tin )

(5)

for k=−N/2, −N/2+1,···, N/2−1. • In the causal case, ha(t) is delayed by NTin/2 and the output is given by y (l ) =

N −1

∑ x(n

l

k =0

− k + N / 2)ha ((k + µl − N / 2)Tin ).

(6)


18

• For a polynomial-based interpolation filter, the impulse response ha(t) is expressed in each interval of length Tin by means of a polynomial as follows: ha (( k + µl ) Tin ) =

M

∑

cˆm ( k )µ l

m

(7)

m =0

for k=−N/2,···, N/2−1. Here the cˆm (k ) ’s are the coefficients and M is the degree of the polynomials. • By using the analog model, the digital implementation structure for polynomial-based interpolation filters can be derived by substituting (7) into (6) giving y (l ) =

N −1

∑

x ( nl − k + N / 2 )

k =0

M

∑

m

cˆm ( k − N / 2)µ l =

m =0

M

∑

v m ( nl ) µ l

m =0

m

(8)

where vm ( nl ) =

N −1

∑ x(n − k + N / 2)cˆ l

m (k

− N / 2)

k =0

(9)

are the output samples of the M+1 FIR filters having the transfer functions Cˆ m ( z ) =

N −1

∑ k =0

cˆm ( k − N / 2) z − k .

(10)


19

Interpretation of the Above Equation • According to Equation (9), the lth output sample y(l) at the time instant t = (nl +µl)Tin can be generated based on the N existing samples x(n) being located at n=nl −N/2+1, nl −N/2+2,···, nl +N/2 in the following three steps: • First, these samples are filtered using M+1 FIR filters having the transfer functions of the form given in (10) • Second, the outputs of these filters, denoted by vm(nl), m are multiplied with constants µ l . • Third, the multiplication results are added, leading to the so-called original Farrow structure shown in Fig. 10. • In this figure, the input to both structures is denoted by x(nl + N/2) to emphasize the fact that this is the last existing sample value when evaluating y(l). • If desired, the transfer functions Cˆ m ( z ) can share the delay elements.


20

x(nl) ^

CM(z) vM(nl)

^

^

C2(z)

^

C1(z)

C0(z)

v1(nl)

v2(nl)

v0(nl)

Hold & sample y(l)

µl Fig. 9. The Farrow structure for interpolation. x(nl + N/2) cM(0)

Fin −1

Z

c1(0)

c0(0)

−1

Z

cM(1)

−1

−1

Z

c1(1)

−1

Z

−1

Z

cM(N − 1)

v M (nl)

c0(1)

Z

c1(N − 1)

v1(nl)

µl

c0(N − 1)

v0(nl)

Fout y(l)

Fig. 10. The Farrow structure.


21

Characteristics of the Farrow Structure • The number of FIR subfilters is M+1. • The length of these subfilters is N. • Filter coefficients are directly determined by the polynomial coefficients of the impulse response ha(t). • The main advantage of the Farrow structure is that all the filter coefficients are fixed. • The only changeable parameters are the fractional interval µl as well as nl that depend on the l th output sampling instant. • During computation, the control of µl is easier than in the implementation based on fixed FD filters. • The resolution of µl is limited only by the precision of the arithmetic, not by the size of the coefficient memory. • These characteristics of the Farrow structure make it a very attractive structure to be implemented using a VLSI circuit or a signal processor. • In fact, the Farrow structure is a general-purpose adjustable FIR filter whose taps are polynomials in a control variable. Any filter property may be varied.


22

POLYNOMIAL-BASED INTERPOLATION FILTERS General approach: How to construct the impulse response ha(t)

• ha(t) nonzero for 0 ≤t 1 is rational, µl becomes periodic, with the periodicity of K. The set of µ values contains K members and can be written as {µ(0)+ k/K | k=0…K-1, 0≤ µ(0) Simpler formulas, simpler programs. BUT: A filter designed using the discrete-time model may not be good for other conversion factors! x(n) Fin

↑K

KFin

FIR filter z(j) Hd(z) KFin

↓L

y(l) Fout=KFin/L

Fig. 25. Model for rational sample rate conversion.


55

Emulating Continuous-Time Signal Processing

• The Farrow structure can be easily generalized to process digitally the reconstructed signal ya(t). • These applications include, among others, interpolating the derivative or integral of ya(t). • Derivative is widely utilized, for example, in finding the location of a local maximum or minimum of a signal. • Integral can be used to calculate the energy of a signal over a given time interval. • We concentrate on determining the derivative of ya(t). Farrow Structure for Determining the Derivative of ya(t)

• In the intervals nTi n ≤t>2. • However, if there are signal components close to Fin/2, i.e., the oversampling factor is small ( high orders feasible

Disadvantages: • Poor response (Lagrange) – Requires an oversampled signal • High order (due to poor response) => Longer delay in causal realisations

References [1] L. Elden, L. Wittmeyer-Koch, and H.B. Nielsen, Introduction to Numerical Computation. Studentlitteratur, Lund, 2004, pp. 107–113. [2] http://mathworld.wolfram.com/UmbralCalculus.html [3] S. Tassart and Ph. Depalle, “Fractional delays using Lagrange interpolators,“ in Proc. Nordic Acoustic Meeting, Helsinki, Finland, 12–14 June, 1996. [4] C. Candan, “An efficient filt ing structure for Lagrange interpolation,” in IEEE Signal Processing Letters, Vol. 14, No. 1, Jan 2007, pp. 17–19. [5] T.J. Goodman, M.F. Aburdene, “Interpolation Using the Discrete Pascal Transform,” Proc. 40th Annual Conf. Information Sciences and Systems, 22–24 March 2006, pp. 1079–1083. [6] Vesa Lehtinen, Markku Renfors, "Structures for Interpolation, Decimation, and Nonuniform Sampling Based on Newton's Interpolation Formula," in Proc. Sampling Theory and Applications (SAMPTA), Marseille, France, 18-22 May 2009. http://hal.archives-ouvertes.fr/hal-00451769/

Vesa Lehtinen

3/3

29.10.2008