JP2900076B2 - Waveform generator - Google Patents

Description

Description: TECHNICAL FIELD [0001] The present invention relates to a waveform generation device used in an electronic musical instrument or the like, and more particularly, to a waveform generation device that generates a waveform using a memory of compressed waveform information.

[Background of the Invention] A PCM-type sound source has a built-in waveform memory for storing musical tones as a sequence of waveform data (PCM data). In operation,
The PCM tone generator generates a tone signal of a desired pitch by reading waveform data from a waveform memory at a speed corresponding to a required pitch.

A drawback of the PCM type sound source is that a memory with a large storage capacity is required as a waveform memory.

In order to avoid this disadvantage, there is a differential PCM method. In the differential PCM method, the number of bits required to represent a difference value between adjacent waveform data is expected to be smaller than the number of bits required to represent waveform data. A data string of difference values (differential data string) is recorded in the. At the time of reproduction, the differential PCM sound source reproduces the waveform data by accumulating the differential data read from the memory of the differential data string.

Unfortunately, the conventional differential PCM method has a problem that the accuracy of reproduction varies depending on the timbre of the original sound.

In the conventional differential PCM recording method, a sequence {x (n)} of waveform data having a certain number of bits (eg, 16 bits) is converted into a sequence {d (n)} of differential data having a smaller number of bits (eg, 8 bits). Convert to The basic principle of the conversion is as follows. First, adjacent waveform data x (k), x (k−
Calculate the difference of 1).

x (k) -x (k-1) = Dx (Equation 1) Here, the difference Dx is difference data represented by 16 bits.
When the size of the difference data Dx is a size that can be expressed by 8 bits (128 to 127), the lower 8 bits of the 16-bit difference data Dx are set as the k-th 8-bit difference data d (k).

(Note that a two's complement is assumed here).

However, if the magnitude of Dx indicates a value that cannot be represented by 8 bits, d (k) is clipped to the maximum value that can be represented by 8 bits.

Clipped to

Note that, in order to reduce the accumulation of errors due to this clip, instead of (Equation 1), Is calculated, and the difference data dx in the 16-bit expression is converted into 8-bit difference data d (k) as described above.

By the above-mentioned clip, the sequence of difference data ｛d
(N) The waveform reproduced from 波形 is distorted. Even more unfortunately, the frequency of occurrence of clips, which is a cause of deterioration of the S / N ratio, depends on the sequence {x (n)} of the waveform data, and therefore the spectrum or timbre of the original sound to be recorded.
As a result, the conventional differential PCM method uses S
The N ratio varied and the desired reproduction accuracy could not be guaranteed.

Therefore, there is a demand for a waveform reproduction technique that can realize a desired waveform reproduction from a memory having a relatively small storage capacity regardless of the type of the original sound.

Further, in a sound source using a waveform information memory such as a waveform data sequence and a difference data sequence, the reproduced waveform data sequence is not output as it is, but is further digitalized after the reproduction process for tone processing or the like. It is often performed to apply a filtering process to a data sequence in a waveform to generate a processed waveform data sequence. This type of sound source (waveform generation device) generally requires high-speed processing. Therefore, when a waveform reproduction technique that can ensure desired reproduction accuracy is applied to this type of waveform generation apparatus, the application does not complicate the configuration of the waveform generation apparatus or reduce the processing speed. It is desired to do so.

SUMMARY OF THE INVENTION Accordingly, it is an object of the present invention to provide a high-quality waveform data sequence that has a simple configuration, can reproduce a high-quality waveform data sequence from a data sequence having a relatively small number of bits regardless of the type of waveform, and performs desired processing on waveform data. It is to provide a waveform generating device that can be applied to a column.

According to the present invention, there is provided a compressed difference data string storage unit for storing a compressed difference data string compressed at a compression rate depending on the magnitude of a fluctuation of a waveform data string representing an audio signal,
A waveform reproducing means for reading out the compressed differential data from the compressed differential data string storing means and reproducing the waveform data; a digital filter means for digitally filtering the waveform data from the waveform reproducing means; Scale factor storage means for storing a scale factor obtained by multiplying an expansion rate for expanding the compressed difference data by a gain coefficient relating to the filtering process of the digital filter means, The means is provided with a multiplying means for multiplying the waveform data reproduced by the waveform reproducing means by the scale factor from the scale factor storage means.

A first feature of the present invention is that the waveform information stored in the storage means as a waveform information memory is obtained by a compressed difference data sequence compressed at a compression rate depending on the magnitude of fluctuation of a waveform data sequence representing an audio signal. Is to be expressed. Thereby, the storage capacity of the waveform information memory (compressed difference data string storage means) is reduced. In order to reproduce the original waveform data sequence from such a compressed differential data sequence, the compressed differential data is accumulated, and the cumulative ratio (compressed reproduced waveform data) is related to the compression ratio by the expansion ratio (reciprocal of the compression ratio). Or a value close to it), whereby a high-quality waveform data string can be reproduced regardless of the type of waveform.

Such a concept of the compression ratio and the expansion ratio does not exist in the conventional differential PCM technology. Conventional difference PCM technology basically takes the difference between adjacent waveform data at the time of recording, records the lower N bits of the difference as difference data, and simply accumulates the difference data at the time of reproduction. This is a method for reproducing waveform data. For this reason, a portion where the difference between adjacent waveform data is large (that is, a portion where the difference is not reflected in the difference data)
Causes fatal distortion and noise.

On the other hand, in the present invention, if the variation of the original waveform data sequence is large, the compressed difference data sequence compressed at the corresponding compression ratio is stored in the compressed difference data sequence storage means. This is reflected in the compressed difference data sequence as faithfully as possible, and high-quality waveform reproduction independent of the type of waveform becomes possible.

On the other hand, in the present invention, it is necessary to multiply the data by the expansion ratio in order to reproduce the original waveform data with the introduction of the expansion ratio which has not existed conventionally. Therefore, if such a waveform reproduction technique is applied as it is to a waveform generation device having a digital filter function such as tone processing, an expansion ratio for waveform reproduction (normalization) is required to generate one waveform data. And the multiplication of the filter gain coefficient for digital filtering must be performed separately, which increases the processing load on the waveform generating apparatus.

Therefore, according to the second aspect of the present invention, a scale factor storage means stores a scale factor corresponding to a value obtained by multiplying an expansion rate and a filter gain coefficient. The multiplication means for multiplying the waveform data by is provided to reduce the number of multiplications required for generating the waveform.

Further, according to the present invention, there is provided a compressed difference data string storage means for storing a compressed difference data string compressed at a compression rate dependent on the magnitude of a fluctuation of a waveform data string representing an audio signal, and A scale factor having a value obtained by multiplying a decompression rate for decompressing the compressed difference data by a filter gain coefficient relating to a digital filtering process to be performed for tone processing of a waveform data string is stored. The reproduction of the waveform data sequence is achieved based on the scale factor storage means and the compressed difference data sequence and the expansion ratio, and the digital filtering process for the waveform data sequence having the filter gain coefficient is achieved. Thus, the compressed difference data string from the compressed difference data string storage means is signal-processed to generate a tone data-processed waveform data string. Waveform generating means, wherein the waveform generating means performs multiplication of the expansion rate and digital filtering by one multiplication using the scale factor from the scale factor storage means as a multiplier. There is provided a waveform generating apparatus having multiplication means for simultaneously performing multiplication of a filter gain coefficient.

Here, the multiplying means is not limited to a specific processing step as long as it performs multiplication at an appropriate processing step of the signal processing of the waveform generating means. The signal processing performed by the waveform generating means can take any appropriate processing order as long as the functions of reproducing and processing (digital filtering) the desired waveform data are achieved.

Embodiment An embodiment of the present invention will be described below with reference to the drawings. First, a waveform recording device will be described, and then a waveform generation device that generates a waveform by using a recording result according to the present invention will be described.

FIG. 1 shows the overall configuration of the waveform recording apparatus. The digital audio tape 1 is a master tape on which a natural musical instrument sound or the like is recorded.
Is transferred to the computer 4 via the. Computer 4
Then, a process described later is performed on the transferred waveform data sequence to generate a compressed difference data sequence and various parameters. The compressed difference data string is written to the PROM 6 by the PROM writer 5. The written PROM 6 becomes the difference waveform memory 11 of the waveform generator (FIG. 4). Further, various parameters (expansion rate and envelope parameter) are used as control information of the waveform generation device.

FIG. 2 is a flowchart of generation of a compressed difference data sequence in the computer 4. First, the parameter a is set in S1, and the waveform data sequence is loaded into the array {x (n)}.
Here, the parameter a is a coefficient used for generating a compression ratio (compression scale factor) described later, and takes a value of a ≧ 1.0. In the next step S2, the waveform data string {x (n)}
That is, the envelope {e (n)} is extracted from x (0), x (1)... X (n). Several methods are available for envelope extraction (for example, Japanese Patent Application No. 61-26).
No. 4205, Japanese Patent Application Nos. 61-264206 and 62-264207). For example, information indicating the size of the waveform data for each predetermined section of the waveform data sequence, for example, the absolute value of the waveform data
The maximum value of | x (n) |, the peak-to-peak magnitude of the waveform data, or the power of the waveform data (for example, the square root of the sum of squares of the waveform data) is obtained as an envelope. As the predetermined section, for example, the basic cycle of the waveform data sequence is selected. In FIG. 3, for each predetermined section X of the waveform data string,
The maximum value of | x (n) | is extracted and obtained as an envelope level ELk.
R _{k is determined} by ER _k = (EL _k −EL _k−1 ) / X, and an envelope value e (n) corresponding to each waveform data d (n) is determined based on these parameters. The extracted envelope {e (n)} is converted into a waveform data string {x
(N) Used to normalize｝. That is, each waveform data x (n) is divided by the envelope value e (n). As a result, a waveform data sequence {x ₀ (n)} from which the envelope has been removed (normalized) is obtained. Eliminating the envelope improves reproducibility in low volume areas.

Next, in S4, in accordance with _{x 1 (n) ← (A} / x 0 max) · x 0 (n), to normalize the waveform data sequence {x ₁ (n)}. Here, x ₀ max is the absolute value of the largest waveform data in the waveform data sequence {x ₀ (n)}. For example, A = A = 3276
If 7.0 is selected, the waveform data sequence {x ₁ (n)} is normalized to ± 15 bits. That is, the maximum value in the waveform data string {x ₁ (n)} is represented by a binary number indicating +32767 or -32767. The reason for the normalization is to make the reproduction levels uniform when reproducing the waveform data sequence {x ₁ (n)} for various waveforms in the reproducing apparatus.

Next, in S5, the difference between each adjacent waveform data of the waveform data sequence {x ₁ (n)} is calculated to form a difference data sequence {d ₁ (n)}, and in S6, the difference data sequence {d ₁ (n) ｛D in ｛
₁ Extract the maximum value dmax of (n)｝. Here is the difference data d
₁ or the number of bits (n) is equal to the number of bits of the waveform data x ₁ (n), or, x ₁ (n) and x ₁ maximum number of bits that can accurately represent the difference _(n-1) to (i.e. d ₁ bit number (n) is the number of bits that can be reproduced completely sequence {d ₁ (n)} the original sequence from _{{x 1 (n)}.} ) next, in S7, Data compression is performed by using x ₂ ← (n) a · (B / dmax) · x ₁ (n) using the compression rate a · (B / dmax). If the number of bits of the compressed difference data d ₂ (n) generated in S8 is 8 bits, B = 12
7.0. Here, if a = 1.0, the waveform data x
₂ The absolute value of the maximum value of the compressed difference value data calculated from (n) is 127.0, and all the compressed difference data can be represented by 8 bits without clipping. That is, B / dm
ax represents the compression ratio without clip. In this case, the reproducibility of the waveform is improved in a portion where the waveform change is sharp, and is deteriorated in a portion where the waveform change is small. The parameter a is a variable for compensating for this point. According to experience, a reproduced sound with the highest sound quality can be obtained within the range of a = 1.0 to 2.0.
It should be noted that the compression ratio a · (B / dmax) depends on the magnitude of the fluctuation of the waveform data sequence. The number of bits of the data of the compressed waveform data sequence {x ₂ (n)} is the same as the number of bits of the waveform data sequence x ₁ (n), and
x ₂ (n) is expressed as a decimal part (for example, the compression ratio a ·
When B / max) is 1/8, the upper 13 bits of data x ₁ (n) are taken as the integer part and the lower 3 bits as the decimal part
x ₂ (n)).

Next, in S8, the compressed waveform data sequence {x ₂ (n)} according to the following equation
From the compressed differential data sequence {d ₂ (n)}.

Here, d ₂ (n) is −B−1 ≦ d ₂ (n) ≦ B, and if the value on the right side of the above equation is smaller than −B-1, d ₂ (n) = − B
Clipped to -1 and d if the value on the right side is greater than B
₂ Clipped to (n) = B. d ₂ (n) is represented by the number of bits determined by the value B, and is 8 bits if B = 127.0.

Note that, instead of S7 and S8, the difference data sequence ｛d
_A compressed difference data sequence {d ₂ (n)} may be obtained from ₁ (n)} using a compression ratio (a · B / dmax).

(D ₂ (n) is d ₂ (n) = − B−1 if the value on the right side is smaller than −B−1, and if the value on the right side is larger than B, as described above.
d ₂ (n) = B clipped).

When a = 1 (no clip), the expression (3) It is equal to x ₂ (n-1). In such a case, S7 and S8
Instead, d ₂ (n) = a · (B / dmax) · d ₁ (n), the compressed difference data sequence {d ₂ (n)} can be obtained.

Next, in S9, the reciprocal dmax / a · B of the compression ratio is stored as the expansion ratio norm in the waveform data reproduction system. As imagined from the above description of the recording system, if the waveform data obtained by accumulating the compressed difference data in the reproduction system is multiplied by the value of norm, the waveform data is thereby obtained regardless of the type of the waveform data. It is standardized to a certain numerical range (for example, -32767 to 32767). An example of the norm value is 5 to 15 for piano waveform data sampled at a sampling frequency of 32 KHz.

In the last S10, the calculated compressed difference data sequence ｛d
₂ (n) Save｝ to a file.

According to the above description, the compressed difference data sequence {d ₂ (n)},
The expansion rate norm and the envelope data sequence {e (n)} were obtained.

The above-mentioned processing (FIG. 2) is performed for some values of the parameter a, data is reproduced, and the data (decompression rate, compressed difference data sequence, envelope sequence) having the highest reproducibility in the auditory experiment is saved, and the compression is performed. The difference data string is written in the PROM 6 to create the difference waveform memory 11.

Next, the waveform generation device will be described. FIG. 4 is a block diagram of an electronic musical instrument having a built-in waveform generator. The microcomputer 8 scans the switch 9 to transfer data corresponding to the selected tone color to the tone generator 10 in a well-known manner, and scans the keyboard 7 to transfer information on a pressed key to the tone generator 10. . According to the present invention, the tone generator 10 calculates the compressed difference data in the difference waveform memory 11 to generate a musical tone waveform W, and emits the tone waveform W through the DAC 12, the amplifier 13, and the speaker 14.

FIG. 5 shows the configuration of the tone generator 10. The interface 24 is for writing data from the microcomputer 8 of FIG. 4 to the tone generator 10, and the envelope level data EL is stored in the envelope data memory 25.
k and the envelope plate data ERk, and scale factors K and B used by the waveform generator 29 are written into the scale factor memory 27. Here, the scale factor K has a value obtained by multiplying the expansion rate norm by a filter gain coefficient.
The scale factor B represents one or more filter coefficients other than the filter gain coefficient used for the digital filtering process in the waveform generator 29. When a temporary IIR digital filter as shown in FIG. 7 described later is used in the waveform generator 29, the scale factor B is one parameter and represents a feedback coefficient. Further, the microcomputer 8 writes frequency data corresponding to the pitch into the address generator 28 via the interface 24. The envelope generator 26 receives the envelope level data ELk and the envelope plate data ERk from the envelope data memory 25.
Then, an envelope E is generated. Address generator
28 is an address A that changes at a speed corresponding to the frequency data
And sends an increment signal i to the waveform generator 29 every time the address A changes. The waveform generator 29 is a waveform generating means for generating a waveform according to the present invention,
In response to the increment signal i from the address generation circuit 28, the tone waveform W is generated by using the compressed difference data D from the difference waveform memory 11, the scale factors K and B from the scale factor memory 27, and the envelope E from the envelope generator 26. Generate

FIG. 6 shows the function of the waveform generator 29. The gate 30 passes the compressed difference data D (for example, expressed by 8 bits) when an increment signal i is given, that is, when the address A changes and the compressed difference data from the difference waveform memory 11 is updated. , And FF32. The accumulator accumulates the compressed difference data D to 8
Generate 16-bit waveform data (cumulative power). This cumulative calculation force W ₀ represents the cumulative value of the compressed difference data, that is, the compressed reproduced waveform data.

Therefore, if the cumulative calculation power is multiplied by the expansion rate norm, reproduced waveform data whose range is standardized to ± 15 bits can be obtained. Further, if the standardized reproduced waveform data is digitally filtered with a filter coefficient such as a filter gain coefficient, waveform data in which a timbre or the like is processed is generated. However, according to this method, the multiplication of the expansion rate for normalization and the multiplication of the filter gain coefficient for the filtering process are performed separately, so that the processing load on the waveform generator 29 increases. For example, the waveform generator 29
Let N be the polyphonic number of this type, this type of multiplication must be performed 2N times per sampling period.

Therefore, in the embodiment, the waveform generator 29 uses a digital filter 33 that performs both normalization and digital filtering.

FIG. 7 illustrates the function of the digital filter 33. Here, a primary IIR filter is used as a digital filter, but the number of orders and filter characteristics are not limited in the present invention. FIG. 7A shows the logical configuration of the LPF in the case of the primary order, and FIG. 7B shows the logical configuration of the HPF. Each of them comprises an input multiplier 35, an adder 36, a delay unit 37, a feedback multiplier 38, and an adder 40.

It should be noted that the input multiplier 35 uses the input data W ₀ as a scale factor K, that is, a scale factor K corresponding to a value obtained by multiplying the expansion rate norm by a filter gain coefficient (K ′). Multiply. This means that two multiplications, W ₀ × norm and K ′ × (W ₀ × norm), are realized in a single multiplication K × W ₀ by hardware multiplication circuits that are individually or shared. ing.

Here, for convenience of explanation, when the normalization process and the digital filtering process are separated, the normalization process is W ₀ × no
The rm is calculated, and the digital filtering is a process using the filter gain coefficient K 'and the feedback coefficient B. The transfer function of the digital filtering process is Against HPF Becomes When Butterworth characteristics are selected, B = (α-1) / (α + 1), K ′ = (1 + B) / 2, α = tan (πfc / fs) (where fc is the cutoff frequency, and fs is the sampling frequency. ).

In electronic musical instruments, it is often desired that the cutoff frequency fc be modulated by an operation input from a performance operator. In such an environment, the microcomputer 8 in FIG. 4 obtains cutoff frequency information from the operation input,
Based on this, a feedback coefficient B and a filter gain coefficient K 'are obtained using a conversion table or the like. The microcomputer 8 stores data of the expansion rate norm,
The microcomputer 8 calculates the scale filter K by multiplying the filter gain coefficient K 'by the expansion rate norm. The K and B obtained in this way are set in the scale factor memory 27 of the tone generator 10 and used by the standardizing and digital filter 33 of the waveform generator 29.

Therefore, by executing the processing shown in FIG. 7, the normalization of the reproduced waveform data by the expansion ratio norm and the digital filtering processing by the filter gain coefficient K ′ and the feedback coefficient B are achieved at a time. . In particular, the multiplication of the expansion rate norm and the filter gain coefficient K '
Is simultaneously achieved by one multiplication shown in 35, and the required number of multiplications can be reduced.

Returning to FIG. 6, the waveform data W ₁ subjected to the tone processing outputted from the digital filter 33 is multiplied by the envelope E from the envelope generator 26 by the multiplier 34 to become the final musical sound waveform output W.

Note that the envelope level and rate data set in the envelope data memory 25 are stored in S2 (second
It can be determined based on the envelope extracted in FIG. Since the number of envelope segments (attack segments, etc.) generated by the envelope generator 26 is generally limited, the extracted envelope should be approximated by a limited number of segment envelopes (envelope level and rate for each segment). become. For such an envelope approximation, for example, a technique disclosed in Japanese Patent Application No. 61-264205 can be used. Basically, an appropriate number of segments is set, and an error between the polygonal linear envelope (for example, a characteristic switching point, ie, a folding point is selected as a point of the extraction envelope) and the extraction envelope at the number of segments is evaluated. Optimum approximation is possible by selecting a polygonal linear envelope that minimizes the error. The result (envelope level and rate information for each segment that defines the selected folded line envelope) is stored in the microcomputer 8 of the electronic musical instrument as reference envelope parameters. In operation, the microcomputer 8
The reference envelope parameter is changed depending on a key touch or the like, and the result is written into the envelope data memory 25.

However, if desired, the envelope may be generated independently of the extracted envelope.

[Modifications] The description of the embodiment is finished above, but various modifications and changes can be made within the scope of the present invention.

For example, the order of processing in the waveform generator 29 is shown in FIG.
The order shown in FIG. 7 is not limited. Therefore, the process of multiplying by the scale factor K can be executed at an appropriate stage. If desired, an envelope corresponding to (E × K) is generated, for example, from the envelope generator 26, and this envelope is multiplied with the data at an appropriate stage, whereby the envelope multiplication by 34 and the scale factor multiplication by 36 are performed. It can be achieved simultaneously with one multiplication operation.

Further, in the above embodiment, the envelope is removed from the waveform in the recording system in order to improve the reproducibility at a low volume level, and the envelope is added to the waveform in the reproducing system. Can be omitted. In the recording system, the envelope is not removed, but in the reproduction system, an envelope may be added.

Also, the expansion ratio norm is not necessarily the reciprocal of the compression ratio, and may take a value close to the reciprocal.

[Effects of the Invention] As described in detail above, in the present invention, a compressed difference data string compressed at a compression rate depending on the magnitude of fluctuation of a waveform data string is stored in a waveform information memory, and this compressed difference data string is stored. A multiplication of an expansion rate (having a value related to the compression rate) for expanding the compressed difference data when reproducing the waveform data stream and generating a processed waveform data stream;
Since the multiplication of the filter gain coefficient in the digital filtering processing for processing the waveform data is simultaneously achieved by one multiplication operation, the burden of the waveform generation processing is reduced regardless of the type of the waveform. High quality waveforms can be generated while reducing the number of waveforms.

[Brief description of the drawings]

FIG. 1 is an overall configuration diagram of a waveform recording apparatus, FIG. 2 is a flowchart of a compressed difference data sequence generation process executed by the computer of FIG. 1, and FIG. 3 is a diagram of an envelope extraction performed in the process of FIG. FIG. 4 illustrates an example of an electronic musical instrument incorporating a waveform generating apparatus according to the present invention. FIG. 5 is a block diagram illustrating a configuration example of a tone generator shown in FIG. 4, and FIG. FIG. 7 is a block diagram of the waveform generator 29 shown in FIG. 7, and FIG. 7 is a block diagram exemplifying the digital filter which also serves as data normalization of FIG. 10 Tone generator 11 Difference waveform memory 27 Scale factor memory 29 Waveform generator 33 Standardized digital filter 35 Multiplier D Compressed difference data norm Expansion rate K Scale factor

Claims (2)

(57) [Claims] 1. A compressed difference data string storage means for storing a compressed difference data string compressed at a compression rate dependent on the magnitude of a fluctuation of a waveform data string representing an audio signal; Waveform reproducing means for reading the difference data and reproducing the waveform data; digital filter means for digitally filtering the waveform data from the waveform reproducing means; and the compression difference data which is an expansion rate related to the compression rate. And a scale factor storage means for storing a scale factor obtained by multiplying an expansion rate for expanding the gain coefficient according to the filtering process of the digital filter means, wherein the digital filter means reproduced by the waveform reproduction means. Multiplying the waveform data by the scale factor from the scale factor storage means Waveform generating apparatus which comprises a multiplying means that. 2. A compressed difference data string storage means for storing a compressed difference data string compressed at a compression rate dependent on the magnitude of a fluctuation of a waveform data string representing an audio signal, and a decompression rate related to the compression rate. A scale factor storing a scale factor having a value obtained by multiplying an expansion ratio for expanding the compressed difference data by a filter gain coefficient relating to digital filtering processing to be performed for tone processing of a waveform data string As storage means, the reproduction of the waveform data sequence is achieved based on the compressed difference data sequence and the expansion rate, and the digital filtering process having the filter gain coefficient for the waveform data sequence is achieved. A waveform that performs signal processing on the compressed difference data string from the compressed difference data string storage unit to generate a tone data processed waveform data string The waveform generating means, wherein the multiplication of the expansion rate and the filter gain according to digital filtering are performed by one multiplication using the scale factor from the scale factor storage means as a multiplier. A waveform generating apparatus comprising multiplying means for simultaneously multiplying coefficients.