Proceedings of the 1998 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP’98), vol. 3,
`pp. 1253–1256, Seattle, Washington, May 12–15, 1998
`
`ENERGY-BASED EFFECTIVE LENGTH OF THE
`IMPULSE RESPONSE OF A RECURSIVE FILTER
`Timo I. Laakso1 and Vesa Välimäki2
`1Helsinki University of Technology, Laboratory of Telecommunications Technology
`P.O. Box 3000, FIN-02015 HUT, Finland
`E-mail: Timo.Laakso@hut.fi
`2Helsinki University of Technology, Laboratory of Acoustics and Audio Signal Processing
`P.O. Box 3000, FIN-02015 HUT, Finland
`E-mail: Vesa.Valimaki@hut.fi, URL: http://www.acoustics.hut.fi/~vpv/
`
`rather heuristic and do not attempt to find an optimal value. In
`[4] it was suggested that the filter be implemented in parallel
`form employing second-order filter sections and using a rough
`time-constant-based measure for the length of the impulse
`response of each section. An upper bound for estimating the
`resulting errors for a given L was derived in [1] and [8] but no
`explicit measure for determining L was given.
`
`ABSTRACT
`A measure for the effective length of the impulse response of a
`stable recursive digital filter based on accumulated energy is
`proposed. A general definition and a simple algorithm for its
`evaluation are introduced, and closed-form expressions are
`derived for first-order IIR filters. The effect of zeros on the
`effective length is analyzed. An upper bound for the effective
`length of higher-order filters is derived using results for low-
`order filters. The new measure finds applications in several fields
`of digital signal processing, including estimation of the extent of
`attack transients for filters with dynamically varying inputs,
`elimination of transients in variable recursive filters, and design
`and implementation of linear-phase IIR systems.
`
`1. INTRODUCTION
`
`The impulse response of a stable recursive digital filter is infi-
`nitely long in principle, but due to exponential decay it eventu-
`ally sinks below the quantization step or the noise in the system.
`Thus, in practice the impulse response of a stable recursive filter
`can be regarded as finite. A measure for the effective length of
`the impulse response of an IIR filter is needed in several applica-
`tions, e.g., in estimation of the effective length of the attack tran-
`sient of a recursive filter [2].
`
`When changing the coefficients of a recursive filter, transients
`will occur. These transients depend on the filter input, but an
`impulse-response-based measure can be used to characterize
`them. A special case of this problem is encountered when the
`transients are eliminated using a novel technique by updating the
`state variables of the filter [10], [11]. The transient can be
`canceled within desired accuracy, but this accuracy depends on
`the effective length of the impulse response of the filter after the
`change of coefficients.
`
`Still another application for the effective length of an infinite
`impulse response is a realization technique for linear-phase IIR
`filters based on cascading a minimum-phase IIR filter H(z) and
`its maximum-phase (unstable or noncausal) counterpart H(z–1)
`[4], [1], [8]. The filtering is based on processing the input signal
`in finite-length blocks of L samples. The basic constraint is to
`choose L so that the impulse response of H(z) has decayed to a
`small enough level. On the other hand, block length L should be
`chosen as small as possible to minimize latency. Although L is an
`essential system parameter, techniques to determine its value are
`
`2
`
`d
`
`2
`( )
`n x n
`=−∞
`
`2
`
` with E
`
`=
`
`x n
`=−∞
`
`(1)
`
`Previously, three different amplitude-based methods have been
`used for measuring the effective length of an infinitely long but
`decaying impulse response. 1) In [7], a general duration d of a
`signal was defined. The discrete-time version of the expression is
`∞∑
`∞∑ ( ) 2
`1=
`E
`
`n
`n
`where E is the total energy of the signal. 2) A traditional tech-
`nique is based on the concept of a time constant. Typically, the
`time constant of the pole with the largest radius rmax is used for
`estimating the decay rate of the impulse response and an ampli-
`tude threshold is chosen to determine the effective length [6].
`Smith has proposed to approximate this time constant as 1/(1 –
`rmax) which is obtained by truncating the Taylor series of the
`exact equation [9], [11]. Based on merely one pole of the system,
`this measure is easy to use but gives a crude estimate for the
`effective length. 3) Furthermore, an amplitude threshold can be
`set and the effective length be determined as the sample index
`where the impulse response ultimately goes below this threshold
`[10]. In principle, this technique gives a better approximation.
`The drawbacks are the lack of analytical methods and the com-
`plication of the measure when the impulse response does not
`decay monotonically.
`
`From the above it is apparent that several ways to measure the
`effective length of infinite impulse responses have been sug-
`gested but none of them seems to have gained wide acceptance.
`This paper introduces a meaningful yet simple and practical defi-
`nition. We define the effective length of the impulse response of a
`general recursive filter based on the accumulated percentage of
`the total energy. This concept has several advantages: 1) the
`energy of an additive disturbance is a natural measure in many
`applications, 2) the total energy of a given filter is easy to deter-
`mine either in the time or in the frequency domain, thanks to
`Parseval’s theorem, and 3) the measure is parametric and thus
`flexible.
`
`Ex. 1025 / Page 1 of 4
`Apple v. Saint Lawrence
`
`

`

`2. EFFECTIVE LENGTH OF A GENERAL
`RECURSIVE FILTER
`2.1 Definitions
`
`Consider an Nth-order recursive filter with transfer function
`−
`−
`+
`+ +
`1
`N
`b
`0
`+
`+ +
`1
`
`
`
`
`
`N
`
`( )
`H z
`
`=
`
`=
`
`( )
`B z
`( )
`A z
`
`(2)
`
`Table 1. Algorithm for computing the effective length of
`a general recursive filter.
`
`Step 1: h n
`( )
`
`=
`
`−
`
`−
`
`k
`
`)
`
`(
`b x n
`k
`0
`
`N
`
`m
`
`Step 0: Compute E and EP for the chosen P. Initialize: n = 0,
`− =
`−
`= =
`−
`=
`
`n= δ , h
`)− =1
`( )
`( )
`(
`
`)1
`(
`)
`...
`(
`2
`)
`0 , E A (
`0
`x n
`h
`h N
`N
`∑
`∑
`−
`(
`a h n m
`m
`=
`=
`1
`k
`) +1
`−
`=
`
`2
`Step 2: E n
`( )
`(
`( )
`h n
`E n
`A
`A
`Step 3: If EA(NP) ≥ EP = PE /100, then NP = n and stop; else n =
`n + 1 and go to Step 1.
`
`)
`
`3. LOW-ORDER ALL-POLE FILTERS
`
`3.1 First-Order All-Pole Filter
`
`b z
`b z
`1
`N
`−
`−
`1
`a z
`a z
`1
`N
`where filter coefficients ak and bk are real-valued (k = 0, 1, ..., N).
`Assuming a stable and causal implementation, the recursive filter
`(2) can also be described via an equivalent difference equation as
`∑
`∑
`−
`
`b x n(
`
`a y n m( ),
`k
`m
`=
`=
`0
`1
`k
`m
`where x(n) and y(n) are the input and output of the filter, respec-
`tively. When the input signal is a unit impulse x(n) = δ(n), which
`equals unity at n = 0 and zero elsewhere, the output y(n) = h(n) is
`the impulse response of the filter.
`
`N
`
`
`
`y n( )
`
`=
`
`N
`
`−
`
`−
`
`k
`
`)
`
`
`
`
`
`nfor
`
`≥
`
`0
`
`(3)
`
`The total energy of the causal impulse response h(n) is defined as
`∑
`∫
`
`2
`
`ω
`j
`
`)
`
`ω
`
`d
`
`(
`H e
`
`ππ
`
`=
`
`1
`π
`2
`
`=∞
`
`n
`
`=
`
`E
`
`(4)
`
`P
`
`=
`
`−
`
`2
`
`P
`
`−
`
`2
`
`Consider a first-order all-pole filter with the transfer function
`−
`=
`−
`1
`(6)
`(1 1
`)
`
`( )
` H z
`az
`where a is real-valued and the pole radius ⎪a⎪ = r < 1 for stabil-
`ity. Its causal impulse response is simply h(n) = an for nonnega-
`tive n. Accumulated energy EA(NP) can be expressed as
`(
`(
`(
`)
`)
`)
`+
`1
`n
`N
`∑ 2
`a
`=
`0
`from which the total energy is also obtained as a limit (NP → ∞)
`as E = 1/(1 – r2). The requirement (5) now becomes
`1
`P
`P
`) ≥
`−
`100
`100
`1
`r
`and the EL can be solved as
`(
`)
`−
`1
`100
`)
`log(
`where the logarithm can have any (positive) base and ⎡·⎤ denotes
`the ceiling operation (i.e., rounding upwards). Note that quanti-
`zation is necessary because NP must be an integer.
`
`1
`
`r
`
`
`
`(7)
`
`⎤ ⎦⎥
`
`r
`
`⎡ ⎣⎢
`
`1
`
`nN
`
`(
`E N
`A
`
`P
`
`) =
`
`(8)
`
`(9)
`
`P(
`E N
`A
`
`=
`
`2
`
`E
`
`
`
`
`
`⎤ ⎥⎥⎥
`
`1
`
`−
`
`/
`2
`
`P r
`
`log
`
`⎡ ⎢⎢⎢
`
`N
`
`P =
`
`Figure 1 presents the EL NP for P = 90%, 95%, and 99% as a
`function of pole radius r computed according to (9). These
`curves show the expected phenomenon that the EL of the
`impulse response increases rapidly as pole radius r approaches
`the value 1. Furthermore, it is seen that the EL is fairly insensi-
`tive to the percentage value so that the lengths corresponding to
`90%-99% energy do not differ much except for when pole radius
`r is larger than 0.9.
`
`3.2 Second-Order All-Pole Filters
`
`Similar derivations can be conducted for second-order all-pole
`filters. Three different cases have to be elaborated separately: a
`complex-conjugate pair, a double real pole, and two distinct real
`poles. The derivations are more involved than in the first-order
`case. Furthermore, exact closed-form formulas cannot be derived,
`but simplified approximations or upper and lower bounds can be
`arrived at for the complex-conjugate case. For the other two
`cases, it is only possible to derive closed-form formulas for
`accumulated energy EA(NP) and total energy E. Unfortunately,
`
`2
`
`−
`1
`z dz
`
`)
`
`− −
`
`1
`
`∫2
`
`( )
`h n
`0
`1
`π
`2
`j
`
`( )
`(
`H z H z
`
`=
`
`where the frequency-domain expression follows from the Parse-
`val relation. The determination of the integral in the z-domain
`has been addressed in [3], for example.
`
`We define the energy-based effective length (EL) as the smallest
`nonnegative integer time index NP by which at least P% of the
`total energy of the impulse response has arrived. The corre-
`sponding accumulated energy EA(NP) can be expressed as
`
`2
`
`( )
`n
`
`≥
`
`=
`
`E
`
`P
`
`E
`
`=∑
`nN
`
`P
`
`=
`
`(
`E N
`A
`
`P
`
`)
`
`P
`h
`100
`0
`Hence, we always require EA(NP) ≥ EP since the effective length
`NP must be an integer. Note that this differs slightly from the
`usual definition of length of the corresponding FIR filter: the
`truncated part contains NP + 1 samples but the effective length
`(5) is one less, NP. The energy-based length (for any percentage)
`of a filter with a unit impulse as the impulse response is thus
`zero, and that of a two-point averager is unity, which is in accor-
`dance with common sense.
`
`(5)
`
`2.2 General Algorithm
`
`The most straightforward way to compute the impulse response
`of a given causal and stable recursive filter is to use the differ-
`ence equation (3). When the total energy E is precomputed, the
`corresponding accumulated energy EA(NP) ≥ EP for the chosen
`percentage P can be determined recursively via the algorithm
`presented in Table 1. This simple algorithm can be used for many
`recursive filters. However, for narrowband filters the length can
`be hundreds of samples. For low-order all-pole filters more prac-
`tical closed-form expressions can be derived.
`
`Ex. 1025 / Page 2 of 4
`
`

`

`The EL can now be solved as
`(
`−
`1
`
`⎡ ⎢⎢⎢
`
`N
`
`P =
`
`where
`
`( , )
`L a b
`
`=
`
`−
`+
`
`(1 2
`ab b
`
`2
`
`)
`
`−
`
`/
`b a
`
`4.2 N Zeros
`
`The conclusions for the first-order filter can readily be general-
`ized for higher-order filters. Consider a general recursive transfer
`function H(z) = B(z)/A(z) with the numerator B(z) of order MB.
`Assuming a fixed denominator, the longest possible impulse
`response corresponds to a delay of MB units (one per each zero)
` of
`and it is attained when the highest-order coefficient bM B
`−
`−
`=
`+
`+ +
`1
`M
`( )
`...
` is large enough compared to
`B z
`b
`b z
`b
`zM
`0
`1
`B
`the others. The EL thus has an upper bound
`{
`}
`( )
`N H z
`P
`
`B
`
`+
`
`M
`
`B
`
`⎫⎬⎭
`
`1
`( )
`A z
`
`⎧⎨⎩
`
`≤
`
`N
`
`P
`
`(15)
`
`The smallest possible EL for the high-order filter is zero which
`naturally occurs due to (approximate) cancellation of all of the
`poles by corresponding zeros. This result is used in the next sec-
`tion to obtain a general bound for high-order filters.
`
`5. HIGH-ORDER RECURSIVE FILTERS
`
`(14)
`
`⎤ ⎥⎥⎥
`
`1
`
`−
`
`+
`2
`
`[
`log ( , )
`L a b
`
`]
`
`)
`
`log
`
`P
`
`/
`
`)
`100
`log(
`a
`(
`)
`2 .
`1
`It is seen that (14) is the same as (9) except for a additive new
`term log[L(a,b)]. Since log(a2) < 0, this term increases the length
`of the impulse response when L(a,b) is smaller than unity, which
`happens when ⏐b – a⏐>⏐a⏐. In the limit the additional term goes
`asymptotically towards the minimum value log[L(a,b)] → log(a2)
`when ⏐b⏐→ ∞, which means that the impulse response is length-
`ened by one sample at most. In this case the numerator approxi-
`mates a unit delay, i.e., 1 – bz–1 ≈ bz–1.
`On the other hand, the impulse response is shorter than (or equal
`to) that without the zero when ⏐b – a⏐<⏐a⏐. For zeros close
`enough to the pole, the EL is suppressed down to zero. When b =
`a, the zero exactly cancels the pole and the impulse response
`reduces to a unit impulse.
`
`0
`
`0.2
`
`0.6
`0.4
`POLE RADIUS
`
`0.8
`
`1
`
`20
`
`18
`
`16
`
`14
`
`12
`
`10
`
`02468
`
`LENGTH IN SAMPLES
`
`Figure 1. The effective length of a first-order all-pole
`filter for P = 90% (solid line), P = 95% (dashed line),
`and P = 99% (dotted line) as a function of pole radius r.
`
`these do not lend to an easy closed-form solution for NP, but they
`can be used to efficiently search for minimum NP by successive
`evaluations. Using binary search, about log2(NP) evaluations are
`needed, as compared to NP steps of the algorithm of Table 1. For
`example, if we can assume that the EL is at most 256, only 8
`evaluations of EA(NP) and E are required. The derivations are
`omitted due to space limitations. Details are available in a long
`version of this work [5].
`
`4. ON THE EFFECT OF ZEROS
`
`The above results consider all-pole filters only. In this section we
`show how the zeros affect the EL of recursive filters’ impulse
`response. A general first-order filter is studied in detail after
`which general conclusions are drawn for higher-order filters.
`
`4.1. General First-Order IIR Filter
`
`Let us consider a first-order IIR filter with transfer function
`−
`−
`=
`−
`−
`1
`1
`(10)
`) (1
`(1
`)
`
`
`( )
`az
`bz
`c
`H z
`where a, b, and c are real-valued and ⏐a⏐<1. The impulse
`response is now
`
`< =
`
`=
`
`Analytical treatment of higher-order filters soon becomes cum-
`bersome. Instead of trying to derive complicated formulas of
`questionable utility, approximate upper bounds are derived. Let
`us focus on the case of effective length for a relatively large P
`(90...99.99%) so that most of the energy has arrived by time
`index NP and we can neglect the tail of the impulse response. We
`define the length-NP truncated impulse response as
`=
`( ),
`for
`0, 1, ...,
`h n
`n
`N
`0
`otherwise
`
`⎧⎨⎩
`
`h
`TR
`
`( )
`n
`
`=
`
`P
`
`(16)
`
`As the truncated impulse response is genuinely finite-length, we
`can obtain a simple approximative limit for the length of the
`convolution of two impulse responses h1(n) and h2(n) with effec-
`tive lengths NP1 and NP2 as
`{
`}
`{
`}
`∗
`≈
`( )
`( )
`( )
`N h n h n
`N h
`n
`1
`2
`1
`P
`P
`TR
`This follows because the length of the convolution of two
`
`≤
`
`+
`
`N
`
`1
`P
`
`N
`
`P
`
`2
`
`(17)
`
`∗
`( )
`n h
`2
`TR
`
`3
`
`(11)
`
`(12)
`
`0 0 1
`
`.
`
`≥
`
`n
`n
`
`n
`
`−
`
`−
`1
`
`n
`
`,
`
`)
`b a
`
`⎧ ⎨⎪ ⎩⎪
`
`0
`c
`
`, ,
`
`(
`c a
`
`( )
`h n
`
`The accumulated energy EA(NP) is (for NP > 0)
`
`⎤ ⎦⎥⎥
`
`−
`
`a
`
`2 2(
`)
`b a
`
`−
`
`)1
`
`n
`
`=∑
`nN
`
`P
`
`(
`1
`
`+
`
`⎡ ⎣⎢⎢
`
`1
`
`(
`E N
`A
`
`P
`
`)
`
`=
`
`2
`
`c
`
`2
`
`−
`−
`−
`+
`=
`2
`2
`) (1
`) (1
`)
`
`
`(
`a
`a
`b
`c a
`c
`from which the total energy is obtained as a limit (as NP → ∞)
`=
`−
`+
`−
`2
`2
`2
`
`1 2(
`
`1) (
`)
`(13)
`E c
`ab b
`a
`
`2
`
`N
`
`P
`
`2
`
`Ex. 1025 / Page 3 of 4
`
`

`

`i.e., P = 100% × (1 – 10–7) = 99.99999%, results in the exact EL
`of NP = 160 samples. Hence, assuming that the energy-based
`criterion is suitable for the application, 20% savings in the proc-
`essing delay can be achieved by using the proposed EL of the
`impulse response.
`
`7. CONCLUSIONS
`
`A new approach for determining the effective length (EL) of the
`impulse response of a recursive filter based on the accumulated
`energy was proposed. The energy-based measure is argued to be
`better suited for many signal processing problems than former
`techniques that focus on the amplitude of the impulse response or
`the time constant of the system. Alongside a simple recursive
`algorithm to determine the EL for any stable IIR filter, closed-
`form formulas were derived for first-order all-pole and pole-zero
`filters. The effect of zeros was studied in a general case, and an
`approximate upper bound was derived for estimating the EL for
`higher-order filters using formulas for low-order filters. The
`results of this paper find applications in several fundamental and
`advanced signal processing problems. An example of the appli-
`cation of the new measure to the design of the block length in
`linear-phase IIR filtering was presented.
`
`8. ACKNOWLEDGMENTS
`
`The authors are grateful to Dr. Jonathan Mackenzie and Dr. Tony
`Wicks for helpful discussions.
`
`9. REFERENCES
`[1] R. Czarnach, “Recursive processing by noncausal digital
`filters,” IEEE Trans. Acoust., Speech, Signal Processing,
`vol. 30, no. 3, pp. 363–370, June 1982.
`[2] R. W. Hamming, Digital Filters. Second Edition. Engle-
`wood Cliffs, NJ: Prentice-Hall, 1983, pp. 244–245.
`[3] E. I. Jury, Theory and Application of the z-Transform
`Method. New York, NY: Wiley, 1964, p. 298.
`[4] J. Kormylo and V. K. Jain, “Two-pass recursive digital
`filter with zero phase shift,” IEEE Trans. Acoust., Speech,
`Signal Processing, vol. 22, pp. 384–387, Oct. 1974.
`[5] T. I. Laakso and V. Välimäki, “Energy-based effective
`length of the impulse response of a recursive filter,”
`unpublished manuscript, Mar. 1997.
`[6] S. J. Orfanidis, Introduction to Signal Processing. Engle-
`wood Cliffs, NJ: Prentice-Hall, 1996.
`[7] A. Papoulis, Signal Analysis. New York, NY: McGraw-
`Hill, 1977, p. 291.
`[8] S. R. Powell and P. M. Chau, “A technique for realizing
`linear phase IIR filters,” IEEE Trans. Signal Processing,
`vol. 39, no. 11, pp. 2425–2435, Nov. 1991.
`[9] J. O. Smith, discussion with V. Välimäki, Nov. 1995.
`[10] V. Välimäki, T. I. Laakso, and J. Mackenzie, “Elimination
`of transients in time-varying allpass fractional delay filters
`with application to digital waveguide modeling,” in Proc.
`Int. Computer Music Conf., Banff, AB, Canada, pp. 327–
`334, Sept. 1995.
`[11] V. Välimäki, Discrete-Time Modeling of Acoustic Tubes
`Using Fractional Delay Filters. Dr. Tech. thesis. Espoo,
`Finland: Helsinki Univ. of Tech., Dec. 1995.
`
`sequences of lengths (NP1 + 1) and (NP2 + 1) is equal to NP + 1 =
`(NP1 + 1) + (NP2 + 1) – 1 = NP1 + NP2 + 1, or NP = NP1 + NP2
`(remember that the effective length is one shorter than the num-
`ber of coefficients!). Applying this result for many convolutions
`we can express a formula for a filter consisting of K subsections:
`{
`}
`∗
`∗
`∗
`≤
`+
`+
`+
`( )
`( )
`( )
`N h n h n
`n
`1
`2
`P
`Let us then consider a transfer function where poles are divided
`into at most second-order real-coefficient sections as follows:
`
`N
`
`1
`P
`
`N
`
`P
`
`2
`
`
`
`N
`
`PK
`
`(18)
`
`
`
`h
`K
`
`( )
`A zk
`
`(19)
`
`kK
`
`=∏ 1
`
`A
`
`( )
`H z
`
`=
`
`( )
`( )
`B z A z
`
`=
`
`( )
`B z
`
`where the numerator B(z) is of order MB, and KA denotes the
`number of sections in the denominator. Combining (18) with
`(15), we obtain an approximative upper bound for the EL as
`{
`}
`∏
`∑
`( )
`N H z
`N
`P
`=
`=
`1
`1
`
`(20)
`
`⎫⎬⎭
`
`1
`( )
`A z
`k
`
`⎧⎨⎩
`
`P
`
`A
`
`kK
`
`≤
`
`+
`
`M
`
`B
`
`⎫⎬⎪ ⎭⎪
`
`( )
`A z
`k
`
`A
`
`kK
`
`⎧⎨⎪ ⎩⎪
`
`( )
`N B z
`P
`
`=
`
`This is a general-purpose result which can be applied to any kind
`of stable filters when the factorization to first or second-order
`real-coefficient sections is available. Note that the obtained esti-
`mate for the EL is an approximate upper bound and it may be
`pessimistic for filters with poles and zeros close to each other.
`
`6. APPLICATION EXAMPLE
`
`Let us then consider a real-life example where the estimation of
`the length of the impulse response of the IIR filter is crucial. As
`discussed in the Introduction, linear-phase IIR filters can be
`implemented by cascading a minimum-phase IIR filter H(z) and
`its maximum-phase counterpart H(z–1). For this the effective
`length of H(z) must be determined. In [4], Kormylo and Jain
`designed a third-order elliptic lowpass filter for the processing of
`a noisy ECG signal. The filter specifications were: passband
`ripple Ap = 0.05 dB, passband cutoff frequency ωp = 0.175π (or
`35 Hz for 400 Hz sampling frequency), and stopband attenuation
`Ap = 16 dB. For the cascaded linear-phase system the ripple val-
`ues are of course doubled, i.e., the composite stopband attenua-
`tion is 32 dB.
`
`For block implementation, an estimate for the length of the
`impulse response of the elliptic filter is required. In [4] it was
`suggested (apparently heuristically) that the length of four times
`the time constant τ of the pole with the largest radius should be
`used, which yields the length estimate of 24.25 sample intervals
`(using Smith’s approximation, i.e., time constant τ = 1/(1 – rmax)
`—in [4] no figures were given). The desired 32 dB stopband
`attenuation suggests that at most 10–3.2 = 0.00063096 or 0.063%
`of the impulse response energy can be lost in the truncation,
`which corresponds to P = 99.937%. This yields an energy-based
`EL (exact, using the algorithm of Table 1) of NP = 21 samples,
`which is not far from the 4τ estimate.
`In [8], Powell and Chau employed a seventh-order elliptic low-
`pass filter with the passband ripple Ap = 0.005 dB, passband
`cutoff frequency ωp = 0.65π and stopband attenuation Ap = 35
`dB. Requiring that a bound for the maximum amplitude of tran-
`sient errors be 70 dB below the signal level, it was derived in [8]
`that the block length of 200 samples is necessary. By requiring
`the residual energy of the impulse response to be below 70 dB,
`
`4
`
`Ex. 1025 / Page 4 of 4
`
`