JP3307483B2 - Digital broadband 90 degree phase shifter - Google Patents

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention
Regarding the phase shifter, Hilbert is an important functional element especially in mobile communication and various advanced signal processing.
t) For accurate and economical implementation of the converter.

[0002] In mobile communications, it is important to effectively use limited radio wave resources, and narrowing the frequency exclusive band used by each communication channel is one of the effective means. The Hilbert transformer uses the S
This is an unavoidable circuit for performing SB modulation and demodulation without causing processing distortion. The Hilbert transformer is a circuit that is indispensable when performing advanced signal processing such as generation of an analysis signal.

[0003]

2. Description of the Related Art A frequency component of an instantaneous spectrum obtained by performing an instantaneous spectrum analysis having a sufficiently high time resolution of about one sampling time is multiplied by -j to a positive frequency component to obtain 90 times clockwise in the frequency domain. The ideal Hilbert transform is performed in the frequency domain by rotating the negative frequency component by 90 degrees in the frequency domain by multiplying the negative frequency component by j.

FIG. 7 shows the characteristics of the Hilbert transformer in the frequency domain. As shown in FIG. 7, the characteristics of the Hilbert transformer in the frequency domain are j and j in the negative frequency domain.
Then, -j is given in the positive frequency domain.

[0005]

(Equation 7)

[0006] The real part R (ω) and the imaginary part X (ω) are converted into a function F using the even function and the odd function as follows.
(Ω).

(Equation 8)

[0007]

(Equation 9)

[0008]

(Equation 10)

[0009]

[Equation 11]

[0010]

(Equation 12)

That is,

(Equation 13) The region of 0 <ω in the continuous system corresponds to the region of 0 <ω <π in the discrete system, and the region of ω <0 in the continuous system corresponds to − in the discrete system.
It is mapped to the area of π <ω <0. Therefore, the characteristic shown in FIG. 7 represents the function {−jsign (ω)}.

[0012]

[Equation 14]

[0013]

(Equation 15)

Equation (11) is a function ｛−jsig in a continuous system.
The response of n (ω)｝ is shown. On the other hand, in the discrete system, the time t is an integer multiple of the sampling interval, and the response to the integer n is considered. That is, the time domain response u (n)
Is

(Equation 16)

[0015]

FIG. 8 shows a conventional SDT.
2 shows an example of a sub-channel arrangement configuration in DFT. $ 0 to $ N / 2 indicate the respective subchannel numbers. When considering a discrete system for finite samples, the time resolution of the Fourier transform F (ω) of f (t) needs to be as high as possible. However, in the ST DFT which is an analysis method for providing an instantaneous spectrum according to the related art, as shown in FIG.
0) straddles the zero frequency and the N / 2th subchannel is π /
It exists over two frequencies (when N is even).

The frequency bandwidth of the sub-channels of the ST DFT is a unit of frequency resolution. Since a detailed structure cannot be obtained and the components in the same sub-channel cannot be separated, Hilbert transform is performed on the frequency bands. I can't do things. Therefore, the remaining ♯1-♯ (N / 2) −1 and ♯ (N) other than the ♯0 and ♯N / 2 sub-channels
/ 2) After performing the frequency domain Hilbert transform on the spectrum components from +1 to ♯N−1, the Short Ti
The approximate signal corresponding to the Hilbert transform signal was synthesized by me IFT.

Although it is an operation that cannot be avoided in the prior art system, by suppressing the # 0 and # N / 2 subchannels as described above, the physical existence of the time-domain signal, that is, the Hilbert transform signal is determined. Since the approximation was performed as much as possible, the operation caused distortion. In order to reduce distortion caused by suppressing the # 0 and # N / 2 sub-channel components, in the related art, it is necessary to set the number N of samples in a frame to be large. In the case of application to communication or the like that requires real-time response, there is an economical problem that an ultra-high-speed processing device is required.

[0018]

[Equation 17]

[0019]

(Equation 18) In the above analysis, it is assumed that N is even, but the case of an odd number is omitted because it can be easily analogized.

[0020]

[Equation 19]

By substituting the calculation result of each of the above sums into equation (16),

(Equation 20)

[0022]

(Equation 21)

[0023]

(Equation 22)

[0024]

(Equation 23) However, when the value of N is finite, it is known that a distortion component is mixed as shown in Expression (18). This distortion is
This is because the components of the sub-channels ♯0 and に お い て N / 2 are suppressed in FIG. So-called F like ST DFT
The amount of delay in the IR type signal processing is required for half of the number of filter stages to satisfy the causal rule. Therefore, there remains a problem that the signals represented by the equations (19) and (20) obtained by removing the distortion of the operation of suppressing the sub-channels ♯0 and ♯N / 2 cannot be obtained within a finite time. become.

[0025]

[MEANS FOR SOLVING THE PROBLEMS] Generalized
DFT (gDFT) has been widely used in the audio analysis domain as a device capable of adjusting the subchannel arrangement as described above. In view of the above problems, an object of the present invention is to apply the above-mentioned gDFT to the ST DFT having a high time resolution, which has been developed for signal processing in a digital transmission system, to provide a degree of freedom in subchannel arrangement.
It is an object of the present invention to provide a digital wide-band 90-degree phase shifter that can remove conversion distortion by using TgDFT and further performing Hilbert conversion of all frequency components.

According to the present invention, as shown in the basic configuration of the present invention in FIG. 1, all the arbitrary signals whose band is limited are composed of ST gDFT analyzing means 1, frequency Hilbert transform means 2, and ST gIFT combining means 3. 9 frequency components
Digital broadband 90 for outputting a signal rotated by 0 degree
A degree phaser is provided.

The ST gDFT analysis means 1 provides an arbitrary signal x (r) (r is a sampling time number, band-limited at a frequency f _s / 2 or less and sampled at a sampling wave number f _s . The sampling time t is t
= Τr, τ is the reciprocal of the sampling frequency f _s and N is the instantaneous spectrum at the sampling time n (n = n′τ) Φ (n) = ｛φ ₀ (n ), φ ₁ (n),…, φ _{(N / 2) -1} (n), φ
The components φ _k (n) of _{(N / 2 + 1)} (n),..., Φ _N-1 (n), φ _N (n)｝ are transformed into a Generalized Short
Determined by t Time DFT (ST gDFT).

That is, when k is 0 < k <N / 2,

(Equation 24) h (*) is a window function of 2 m frames satisfying the conditions h (0) = 1 and h (qN) = 0, where q is an arbitrary integer, N
Is a natural number that is an even number of samples in the frame, and k is N / 2
If <k < N,

(Equation 25) The instantaneous spectrum φ (n) having the component as is obtained.

The frequency Hilbert transform means 2 converts the obtained instantaneous spectrum Φ (n) into a frequency domain Hilbert transform, that is, if k is 0 < k <N / 2, which corresponds to the positive frequency of the instantaneous spectrum. The real part of the instantaneous spectrum component φ _k (n) is a _k (n) and the imaginary part is b _k (n) (the instantaneous spectrum component is φ _k (n) = a _k (n)
+ Jb _k (n)), the value obtained by inverting the sign of the real part a _k (n) is defined as the imaginary part, and the value of the imaginary part b _k (n) is defined as the real part.

(Equation 26) Ask for.

Also, k corresponding to a negative frequency is N /
When 2 <k < N, the value of the real part a _k (n) of the component of the instantaneous spectrum is defined as the imaginary part, and the value of the imaginary part b _k (n) is defined as the real part.

[Equation 27]

[0031]

[Equation 28]

According to the present invention, the instantaneous spectrum obtained by the ST gDFT analysis means 1 is obtained by combining the frequency Hilbert transform means 2 and the ST gIFT synthesis means 3.

(Equation 29)

Further, according to the present invention, the ST gD
FT analyzer 1, transversal to the frequency Hilbert transform unit 2, and the processing of the ST GIFT synthesizing means 3, the coefficient values of the unit impulse unit impulse response u obtained by executing against _s (n) The filter provides a digital wideband 90 degree phaser which executes by outputting a signal obtained by multiplying the sum of the input signal and the filter coefficient by an appropriate constant.

[0034]

As shown in FIG. 2, the sub-channel arrangement of the ST gDFT is such that, for a positive real number サ ブ, sub-channels existing in the positive frequency domain are shifted to the right by ξ and to the left in the negative frequency domain. You. Its instantaneous spectrum φ
_k (n) is given by the following equation (21).

[Equation 30] Although N is assumed to be even, the case where N is odd can be easily estimated by analogy, and a description thereof will be omitted.

Equation (21) is a definition of ST gDFT, and the window function h (*) is equal to the definition of ST DFT of equation (14). Also, the inverse transform ST g of ST gDFT
IFT is defined as follows.

(Equation 31)

Equation (22) can be summarized as follows.

(Equation 32)

The instantaneous spectrum components φ _k (n) and φ _Nk (n) in equation (23) have a complex conjugate relationship with each other as shown below.

[Equation 33] Since ST (gIFT) y (n) is given by the sum of complex conjugates as shown in equation (23), it is clear that it converges to a real number.

By directly substituting the instantaneous spectrum φ _k (n) defined by ST gDFT in equation (24) into equation (23), the signal y (n) can be calculated as follows.

(Equation 34)

When r−n ≠ qN,

(Equation 35) Becomes Therefore, the signal y (n) is obtained as follows.

[0040]

[Equation 36] Since the sin function always becomes 0 at an integral multiple of the angle π radian, y (n) = 0 holds.

Therefore, it is understood that the signal y (n) always converges to x (n) if a constraint for limiting ξ to １ ／ is newly set. This constraint ξ = １ ／, as shown in FIG.
the lower fringe of the subchannel ♯0 shifted by ♯ and ♯
The upper fringe of N-1 touches the zero frequency and ΔN /
An upper fringe of 2-1 and a lower fringe of ♯ (N / 2) +1 touch the π frequency, which is the shifted sub-channels {0♯ {N / 2) −1 and ♯ (N / 2) + 1♯. N-1
Corresponds to the case where the entire frequency range is covered.

Further, regardless of the value of N, ST gDF
In T, by restricting ξ = ξ, ST DF
In T, the zero frequency suppression in which only the limit state of N → ∞ is allowed is realized, and the Hilbert transform {−j sign ω} in the Fourier transform region can be realized without distortion.

[0043]

FIG. 3 shows a digital broadband 90 according to the present invention.
1 shows a first embodiment (corresponding to claim 1) of a phase shifter. In FIG. 3, reference numerals 11 to 22 indicate multipliers, and reference numerals 27 to 29 indicate adders.
Reference numerals 24 and 26 denote a shift register (SFR) that forms a delay element of 2 mN (number of frames 2 m, number of samples in a frame N) stages, and reference numeral 25.
Is a coefficient ROM in which weight coefficients of the window function (h (n)) respectively corresponding to the 2mN delay elements are written. Note that the above reference numerals are attached only to one circuit block shown in the preceding stage for simplification of the drawing,
As shown in FIG. 3, all the actual circuits are used for N / 2 circuit blocks except for the adder 29. All or a part of the operation of each of these circuits or circuit blocks may be realized by software.

Before explaining the circuit operation shown in FIG.
First, mathematical verification of the first embodiment is performed. ST gDF
The instantaneous spectrum obtained by exchanging the real part R _k (n) of the instantaneous spectrum φ _k (n) obtained by limiting the parameter の of T to １ ／ and the imaginary part X _k (n) whose sign is inverted is represented by ST
The signal g (n) obtained by gIFT is the input signal X (n).

(37)

ST gDFT spectrum φ _k (n) =
By substituting R _k (n) −jX _k (n) into equation (28) specifically,

(38)

From the above equations (28) and (29), FIG.
The structure of the first embodiment of the present invention shown in FIG. That is, the spectrum of ST gDFT φ _k (n)
= R _k (n) −jX _k (n) real part R _k (n) and imaginary part X _k
The value of (n) is determined by the inner product of the input signal x (r) and each carrier coefficient, and the window function h (n), as shown in the braces before each of the first and second terms on the right side of Equation (29). ) And convolution operation. Therefore, they can be implemented in a transversal filter (FIR) configuration, where the input signal x
Multiplier 11, 1 for multiplying (r) by the coefficient
2, a coefficient ROM 25 in which a weight coefficient of the window function h (n) is written, two 2mN-stage shift registers 24 and 26 corresponding to each operation of a real part and an imaginary part, and each of the filter coefficients and the shift register. Each of the multiplier groups 13 to 16 and 17 to 20 for multiplying the delayed output, and each operation result of each of the multiplier groups is added to output (R _k (n), X _k
(n)).

The real part R _k (n) and the imaginary part X _k (n), which have been added and output, are subjected to frequency Hilbert transform, so that the real part and the imaginary part are exchanged as shown in FIG. Finally, for ST gIFT, each carrier coefficient outside the curly braces of the first and second terms on the right side of equation (29) is
The corresponding operation results are multiplied by 1 and 22, and they are added and output by the adder 29. As described above, the adder 29 outputs the result of adding the N / 2 operation blocks.

Next, it will be verified that the above equation (29) correctly executes the Hilbert transform. By rearranging the right side of equation (29) using a complex exponential function,

[Equation 39]

Here, if the second term on the right side is rewritten using the cyclic property of the complex exponential function,

(Equation 40)

If the parameter N−k of the second term on the right side is k and the sum is re-expressed,

[Equation 41] It proves that the Hilbert transform is executed correctly.

FIG. 4 shows a digital broadband 9 according to the invention.
9 shows a second embodiment (corresponding to claim 2) of a 0-degree phase shifter. 4 that are the same as those in FIG. 3 described above are given the same reference numerals, and will not be described again here. A major difference between FIG. 4 and FIG. 3 described above is that each of the multipliers 11, 1 in the input stage as shown in FIG.
2 is that the respective carrier coefficients of FIG. This means that the above-described operation of the frequency Hilbert transform unit in the intermediate part in FIG. 3 is processed on the input side, and therefore, the real part R _k (n) and the imaginary part X _k (n ) Are excluded from each other.

According to the second embodiment, the real part R _k (n) of the instantaneous spectrum φ _k (n) obtained by limiting the parameter の of the ST gDFT to １ ／ and the imaginary part X whose sign is inverted _k
(n) is replaced, and instead of obtaining the instantaneous spectrum subjected to the Hilbert transform, the definition equation (21) of the STgDFT is used.
The instantaneous spectrum of the Hilbert transform is directly obtained by replacing the operator of

(Equation 42)

The signal g (n) obtained by ST gIFT of the obtained Hilbert-transformed instantaneous spectrum is

[Equation 43] Since the circuit configuration of FIG. 4 is directly derived from the above equation (34) in the same manner as described with reference to FIG. 3, it will not be described again here.

FIG. 5 shows a digital broadband 9 according to the invention.
9 shows a third embodiment (corresponding to claim 3) of a 0-degree phase shifter. FIG. 5 shows a shift register (SFR) 31,
Coefficient ROM 37, multipliers 32-35, and adder 36
2 shows a configuration of a general transversal filter consisting of: The said coefficients ROM, and writes the unit sample response values u _s (n) is, according to this configuration, instead of the weight of the window function of FIG. 3 and FIG. 4 described above, the unit sample response values u _s ( n) is used. This allows
With this configuration alone, all the processing of the present invention is executed in exactly the same manner as described with reference to FIGS. The mathematical verification will be described below.

[0055]

[Equation 44]

The terms shown in curly braces are simplified. The two terms are in a complex conjugate relationship with each other, and are as follows.

[Equation 45]

Therefore,

[Equation 46]

The unit sample response u _s (n) is obtained by setting the input signal x (r) to 1 only at time 0 in equation (37),
It is obtained as the output g (n) when all others are set to 0.
That is,

[Equation 47] In ST gDFT, if the window function h (*) is an ideal Nyquist function, an ideal instantaneous spectrum can be obtained. Therefore, the infinite-length Nyquist function is given by the equation (3)
Substituting 8), an ideal unit sample response u _s (n) is obtained. fact,

[Equation 48] Becomes Thus, it has become clear that the ST gDFT Hilbert transform gives an ideal response if attention is paid to the instantaneous spectrum analysis, that is, the window function of the ST gDFT. At the same time, u _s (n) has ST gDFT
Is a feature that the parameter N indicating the frame length is not included.

Next, an example of a method for determining a window function will be described. If the infinite Nyquist function is used, ST gDF
The T Hilbert transform shows an ideal response. However, if the processing delay is finite, the Nyquist function must be truncated to a finite length. However, truncation to a finite length causes Gibbs phenomenon and induces processing distortion in the band.
In order to suppress this non-linear phenomenon, Kaiser (Kaize)
r) The Nyquist function is truncated to a finite length using the smoothing function K (β, n). The same applies to other smoothing functions, which can be easily inferred, and a description thereof will be omitted.

H (n) = N (n) K (β, n) (40) where N (n) is an infinite Nyquist,

[Equation 49] And K (β, n) is

[Equation 50] Where I ₀ (*) is a Bessel function of one kind depending on the deformation 0, β is a positive real number that changes the degree of energy concentration on the main lobe, and a Kaiser smoothing function.

Therefore, from the above equations (38) and (40), applying the Kaiser smoothing function,

(Equation 51)

[0062]

(Equation 52) Therefore, when the Kaiser smoothing function is used, the weight coefficient of the above equation (41) may be written in the coefficient ROM 37 of FIG.

FIG. 6 shows a digital broadband 9 according to the invention.
14 shows a fourth embodiment of the 0-degree phase shifter. In FIG. 6, the input buffer rotator 41 (buffer size 2 mN;
(N is the frame length, 2m is the number of frames), and the input data x (r) is sequentially and cyclically written in each sampling cycle. Wind rotator 42 (window length = 2 mN)
Stores the weight coefficient at each timing of the window function h (n). Other functional elements, multipliers 43 to 4
7 and adders 48 to 51 are ordinary arithmetic elements. However, accumulators (Acc) for performing cumulative addition of inputs are used in the adders 48 and 49. The two rotators 41 and 42 have a sampling period × 2 mN
Rotate in synchronism with each other in each cycle. However, for the input buffer rotator 41, a new signal x
Each time (n) is input, the oldest input data is rewritten thereby, and that position becomes the start position of the next rotation. Therefore, the data in the input buffer is shifted (delayed) by one stage, and the multiplier 43
In addition, the product-sum operation of the input x (n) and the coefficient of the window function h (*) performed by the accumulators 48 and 49 executes a convolution operation similarly to the transversal filter. At the end of the operation after one round, the accumulator 4
8, 49 are cleared to prepare for the next operation. Note that the rotation is for functionally explaining the calculation operation of the present embodiment, and it is needless to say that a predetermined address such as a RAM or a ROM may be cyclically accessed at a constant period.

This embodiment is directly derived from the equation (29), as in the case of FIG. A major feature of this embodiment is that, in FIG. 3, the operations in the curly braces of the first and second terms on the right side of Expression (29) are performed separately by each transversal filter. This embodiment is characterized in that the above operations are common, and the number of times of multiplication is further reduced, thereby achieving high-speed operation. That is, h (n−r) x (r) common to the first and second terms is obtained by first multiplying the data from the two rotors 41 and 42. After that, each carrier coefficient a, b corresponds to each of the first and second terms.
, And sequentially accumulate the results.
Is added to. By this operation, the first and second terms on the right side of Expression (29) are obtained. As described above, the number of times of multiplication per sample is significantly reduced by the common use of the convolution operation as compared with the configuration using two transversal filters shown in FIG. 3, and high-speed operation becomes possible.

In FIG. 6, the circuit configuration other than the above is substantially the same as that of FIG. 3 even if there is a difference in the arrangement position of the coefficient 2 / N of the output stage, so that it will not be described again here. Note that the circuit configuration of the ST gDFT analysis unit shown in FIG.
The above-described embodiment of FIG.
Needless to say, the present invention can be applied similarly to the case of

[0066]

As described above, according to the present invention,
By providing a new constraint in ST gDFT that limits ξ to １ ／, the signal y (n) always converges to x (n), whereby the lower fringe of subchannel ♯0 shifted by ♯ and ♯N− The upper limit fringe of 1 contacts the 0 frequency, and the upper limit fringe of ♯ (N / 2) −1 and ♯ (N /
2) The lower fringe of +1 touches the π frequency. Therefore, the shifted sub-channels # 0 to (N / 2) -1 and ♯ (N / 2) +1 to ♯N-1 can accurately cover the entire frequency range.

Further, according to the present invention, by restricting ξ = １ ／ in ST gDFT, zero frequency suppression, which is only allowed in the extreme state of N → ∞ in ST DFT, is realized irrespective of the value of N Thus, the Hilbert transform {−j sign ω} in the Fourier transform region can be realized without distortion.

Further, according to the present invention, as described above, the digital wideband 90-degree phase shifter can be realized with a very simple circuit configuration and high performance.

[Brief description of the drawings]

FIG. 1 is a block diagram showing a basic configuration of a digital wideband 90-degree phase shifter according to the present invention.

FIG. 2 is a diagram illustrating an example of an ST gDFT subchannel arrangement according to the present invention.

FIG. 3 is a circuit block diagram showing a first embodiment of a digital wideband 90-degree phase shifter according to the present invention.

FIG. 4 is a circuit block diagram showing a second embodiment of the digital wideband 90-degree phase shifter according to the present invention.

FIG. 5 is a circuit block diagram showing a third embodiment of the digital wideband 90-degree phase shifter according to the present invention.

FIG. 6 is a circuit block diagram showing a fourth embodiment of a digital wideband 90-degree phase shifter according to the present invention.

FIG. 7 is a diagram illustrating characteristics of a Hilbert transformer in a frequency domain.

FIG. 8 is a diagram showing an example of a conventional ST DFT subchannel arrangement.

[Explanation of symbols]

 DESCRIPTION OF SYMBOLS 1 ... ST gDFT analysis T gDFT analysis means 2 ... Frequency Hilbert transformation means 3 ... ST gIFT synthesis means 24, 26, 31 ... Shift register 25, 37 ... Coefficient ROM 41 ... Input buffer rotator 42 ... Window rotator 48, 49 ... accumulator

────────────────────────────────────────────────── ─── Continuation of the front page (56) References JP-A-4-135308 (JP, A) JP-A-1-157611 (JP, A) JP-A-1-128609 (JP, A) Takashi Ishiguro, Norihiro Hattori , Kozaki Yasunari, Kishi Masanori, Application of Generalized Short Time DFT to Hilbert converter, Proceedings of the 1994 IEICE Spring Conference, Pt. 3, p. 3-314 (58) Field surveyed (Int. Cl. ⁷ , DB name) H03H 17/08 G06F 17/14 H04B 1/66 H04B 14/04 H04L 27/32

Claims (3)

(57) [Claims] 1. A digital wide-band 90-degree phase shifter that outputs a signal obtained by rotating all frequency components of a band-limited arbitrary signal by 90 degrees is band-limited at a frequency f _s / 2 or less, and has a sampling frequency f _s . any signal x (r) (r being sampled indicates the number of the sampling time, the sampling time t is t = .tau.r, tau is the reciprocal of the sampling frequency f _s,
N is a natural number that is an even number of samples in the frame)
Instantaneous spectrum at sampling time n (n = n′τ) The component φ _k (n) of Generalized Sho
By t Time DFT (ST gDFT), that is, when k is 0 < k <N / 2, h (*) is a window function of 2 m number of frames satisfying the conditions h (0) = 1 and h (qN) = 0, where q is an arbitrary integer, N is a natural number that is an even number of samples in the frame, and k is If N / 2 <k < N, then: G to find the instantaneous spectrum φ (n) with
DFT analysis means: The obtained instantaneous spectrum Φ (n) is converted into a frequency domain Hilbert transform, that is, if k is 0 < k <N / 2, which corresponds to the positive frequency of the instantaneous spectrum, the instantaneous spectrum component φ _k ( The real part of n) is a _k (n) and the imaginary part is b _k (n)
(The instantaneous spectrum component has φ _k (n) = a in the real part and the imaginary part.
_k (n) + jb _k (n)), the value obtained by inverting the sign of the real part a _k (n) is defined as the imaginary part, and the value of the imaginary part b _k (n) is defined as the real part. 4] When k is N / 2 <k < N corresponding to a negative frequency, the value of the real part a _k (n) of the component of the instantaneous spectrum is defined as the imaginary part, and the value of the imaginary part b _k (n) is determined. Instantaneous spectrum component with the real part as 2. Combining the frequency Hilbert transforming means and the ST gIFT combining means,
Instantaneous spectrum obtained by T analysis means 3. The processing of the ST gDFT analysis means, the frequency Hilbert transform means, and the ST gIFT uses a value of a unit impulse response u _s (n) obtained by executing the unit impulse as a coefficient. The digital wideband 90-degree phase shifter according to claim 1, wherein the transversal filter is executed by outputting a signal obtained by multiplying a value obtained by calculating a sum of an input signal and a filter coefficient by an appropriate constant.