JPH0678010A - Method and system for restoration of timing - Google Patents

Abstract

PURPOSE: To provide a device and a method for receiving a continuous time pulse amplitude-modulated pass-band signal including local clock synchronous technology. CONSTITUTION: A pulse-amplitude modulated signal is sampled periodically with a lock clock 58 in order to generate received samples. In order to lead out the corrected samples, convolution and impulse response functions of the received samples are evaluated with a specific value of a phase difference variable. A filter 68 with a narrow transmission band is used and a periodic signal is led out from the corrected samples. This periodic signal has a period equal to a symbol period of a pulse amplitude modulated signal. A series of the samples in each period of the periodic signal is compared so as to discriminate whether the corrected samples are delayed behind or preceded to the optimum samples. When the corrected samples are delayed behind or preceded to the optimum samples, the phase difference variable used for evaluating the convolution is changed by fixed discrete step size.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION The present invention relates to pulse amplitude modulation (PA
M) is used for a timing recovery method and system for a communication system.

[0002]

BACKGROUND OF THE INVENTION In a typical communication system receiver, an analog-to-digital converter is used to convert the received continuous time signal into a discrete time format. One problem encountered in this type of system is that the local receiver clock and the remote transmit clock are asynchronous. If the receive clock is slower than the transmit clock, then one sample of the received continuous time signal is lost after a sufficiently long period of time. On the other hand, if the local receiver clock is faster than the remote transmit clock, then an extra sample of the received continuous time signal is obtained after a sufficiently long time. The problem of synchronizing the local receiver clock to the remote transmit clock is an important problem in many telecommunications systems. The present invention solves this problem in a communication system utilizing pulse amplitude modulation.

In order to facilitate understanding of the present invention, PA
M will be described in some detail. First, the baseband PAM will be described, and then the passband PAM will be described. Baseband P
AM is commonly used in metallic media such as twisted wire pairs, where the signal spectrum has zero frequency (DC).
Is allowed to extend downwards.

The PAM system is extended to passband transmission by introducing a sinusoidal carrier signal. Passband PAMs are commonly used with transmission media that have a highly constrained bandwidth, such as radio. It uses two sinusoidal carrier signals that have the same frequency but a 90 degree phase difference and is modulated by the real and imaginary parts of the complex-valued baseband signal. In the special case of passbands, the commonly used phase shift keying (PS
K), amplitude and phase modulation (AM-PM) and quadrature amplitude modulation (QAM). For example, Edward A. Lee and David G. See Messerschmitt, "Digital Communications," 1988, Kluwer Academic Press, Boston (henceforth "Lee et al."), Page 146.

A baseband communication system is shown in FIG. 1 (see Lee et al., Page 148). The baseband system 10 of FIG. 1 includes a transmitter 12 and a receiver 14. Transmitter 1
At 2, the incoming stream of bits B _k is encoded by the encoder 16 into a stream of symbols A _K. Bit is "0"
Alternatively, the symbol estimates a value from the given alphabet of the symbol, while it can simply take the value of "1". This alphabet is a set of symbols used for transmission. The baseband signal has a real-valued alphabet that is just a list of real numbers, eg, A _k is the alphabet {-3, -1,
The value starts from +1, +3}. The encoder 16 in the baseband transmitter 12 is, for example, a set {00, 01, 10, 1
A pair of bits from 1} to the alphabet {-3,-
Map to one of the four levels from 1,1,3}.

Since encoder 16 maps many bits into a single data symbol, there is a distinction between "symbol rate" and "bit rate". The "symbol rate" is also referred to as the "baud rate". For example, if the encoder maps two bits to each symbol, the symbol rate is half the bit rate. After being generated by the encoder 16, the symbol A _k
Is applied to the transmit filter 18. The transmit filter 18 produces a continuous time signal s (t) for transmission on a continuous time channel 20.

Impulse response g of the transmission filter 18
(T) is called a pulse waveform. The output s (t) of the transmission filter 18 is a pulse waveform and a symbol sequence.

[0008]

[Equation 1]

Is the convolution of f _b = 1 / T
_b is the baud rate. This signal can be interpreted as a series of possible overlapped pulses with the amplitude of each pulse determined by the symbol. Such a signal is called a pulse amplitude modulation (PAM) signal regardless of the pulse waveform. Baseband PAM and its generalization to passband are among the most common signaling methods. QAM, PSK, BPS
Various technologies, including K, PRK, QPSK, DPSK and AM-PM are special cases of PAM. Lee et al., 14
9-150. At receiver 14, channel 20
The signal R (T) received via the signal is processed by the receiver filter 21 with the impulse response f (t) to produce the output signal Q (t). Received signal R
(T) is also processed by the timing recovery circuit 22 which recovers the clock for use by the sampler 24.
The signal Q (t) is sampled by the sampler circuit 24. The sample Q _k output from the sampler 24 is the reconstructed symbol

[0010]

[Equation 2]

Processed by the decision unit 26 to form Then the symbol is the reconstructed bit stream

[0012]

[Equation 3]

Decoded by decoder 28 to form There are few practical communication channels that can transmit baseband signals. Most physical transmission media are incapable of transmitting DC or near DC frequencies contained in baseband signals. Passband strategies are considered here. The passband policy is considered as using a discrete time signal representation.

The passband strategy is to construct a signal s (nT) where s (nT) is a discrete time analog to s (t) and 1 / T is the sampling rate. In this case the signal s (nT) is complex and contains information in both its real and imaginary parts. The signal s (nT) is modulated to obtain a signal z (nT) modulated with a complex-valued carrier. A transmitter 30 that accomplishes this is shown schematically in FIG. 2 (see Lee et al., Page 170). Transmitter 30
Where the bits b _k are processed by the encoder 32 to produce the complex symbol a _k = a (kT) where f _s = 1 / T is the sampling frequency. The passband PAM signal is a list of complex numbers, eg {-1, -j, +1, +.
j}, which is the alphabet. For alphabets with M = 4 symbols, each symbol may represent log ₂ M = 2 bits. Complex-valued alphabets are best represented by plotting the alphabet as a set of points in the complex plane. Such a plot is called a signal constellation. Two common constellation maps are illustrated in Figures 3A and 3B. Returning now to FIG. 2, the complex symbol a (kT) is processed by the transmit filter 34. In discrete time display, the transmit filter 3 of FIG.
4 has a transfer function h (nT). The output of the transmission filter 34 is a complex baseband signal.

[0015]

[Equation 4]

[0016] Complex baseband signal s (n
T) is then multiplied by the complex-valued carrier in modulator 38:

[0017]

[Equation 5]

Where ω _c is the carrier frequency. The modulator 38 is thus a complex number modulated signal.

[0019]

[Equation 6]

Is output. The signal z (nT) is complex-valued and therefore cannot be transmitted on a real-valued channel. However, all signaling information is contained in the real part of the signal, which can be transmitted on the realised channel.
Therefore, the circuit 39

[0021]

[Equation 7]

Used to obtain this

[0023]

[Equation 8]

The coefficient ensures that the power in x (nT) is the same as the power in s (nT). The timing in the discrete time transmitter 30 of FIG. 2 is controlled by the transmitter clock (not shown). The signal x (nT) is processed by a digital-to-analog converter 40 for conversion into a continuous time signal x (t) for transmission to a remote receiver via a transmission channel 41. It should be noted that the function z (nT) is analytic, ie its Fourier transform does not contain negative frequency components.

The signal x (nT) is

[0026]

[Equation 9]

Is displayed as. This is the signal x (nT)
Are carrier signals cos (ω _c nT) and −sin (ω _c n
T) two real-valued baseband PAM signals, each modulated by

[0028]

[Equation 10]

Is equal to Since these two carriers are 90 degrees out of phase with each other, they are said to be quadrature, where cos (ω _c nT) is said to be the in-phase component and -sin (ω _c nT) is said to be the quadrature component. FIG. 4 shows another embodiment for the transmitter shown in FIG. 2 (see Lee et al., Page 172). In the transmitter 50 of FIG. 4, the bit stream b _k is the real symbol stream Re { _ak } and Im { _ak }.
Are processed by the encoder 51 to form Each symbol stream has a first and a second baseband signal s _r (nT).
And the transfer function to form s _i (nT)

[0030]

[Equation 11]

Processed by the transmit filter 52 having The first baseband signal is then multiplied by carrier cos (ω _c nT) in modulator 53, and the second baseband signal is then modulated by carrier si in modulator 54.
It is multiplied by nω _c nT. This signal is then combined using subtraction element 55 to form the passband signal x (nT). This signal is converted into an analog signal x (t) by the digital-analog converter 56.

In the illustrated example, the sample rate is f _s
= 1 / T = 7200 Hz. Symbol rate f _b = 1 /
T _b is, for example, 2400 symbols per second. The bit rate is 9600 bits / second or 4800 bits / second.
The carrier frequency f _c = ω _c / 2π is, for example, 1800 Hz
Is.

[0033]

The object of the present invention is to obtain the signal x after it has been transmitted to the remote station via the transmission channel.
To provide a receiver for receiving (t). In particular,
It is an object to provide such a receiver that operates using digital signal processing. In such a case, it is necessary to sample the continuous time signal transmitted to the receiver over the channel. The receiver must include some technique for local clock synchronization because the local clock that samples the received signal at the receiver is not synchronized with the transmit clock. It is another object of the present invention to provide such a local clock synchronization technique.

[0034]

According to an embodiment of the present invention, a receiver receiving a continuous time pulse amplitude modulation (PAM) passband signal uses a local clock to generate the received samples. And an analog-to-digital converter for sampling the continuous-time PAM passband signal. The received samples are then corrected for the asynchrony between the local receiver clock and the remote transmit clock. In other words, the timing utilized at the transmitter is locally restored at the receiver. To recover the transmitter timing at the receiver, a timing adjustment filter evaluates the convolution of the received sample and impulse response functions obtained using the local clock with a particular value of the phase difference variable.
The timing adjustment filter outputs the corrected sample. The corrected sample is the optimum sample when the corrected sample is obtained with the optimum value of the phase difference variable. At the optimum value of the phase difference variable, there is synchronization between the local and remote clocks.

The value of the phase difference variable used to evaluate the convolution is obtained using a feedback technique as follows. A periodic signal is derived from the corrected samples generated by the timing adjustment filter. This signal is by way of example a sinusoidal signal at the symbol rate. Starting from the initial value of the phase difference variable, the timing difference detector compares a series of samples of the periodic signal during each period of the periodic signal, with the corrected sample being ahead or behind. Determine if it is the best sample. The phase difference variable is increased or decreased by a constant discrete step size in each period of the periodic signal until the optimum value of the phase difference variable is reached, and the corrected sample is the optimum sample. In this way, local and remote clock synchronization is achieved.

[0036]

DESCRIPTION OF THE PREFERRED EMBODIMENT A receiver 60 according to the present invention is shown in FIG. The continuous time signal x (t) generated by the transmitter (see FIGS. 3 and 5) and transmitted by the channel 59 having a particular transfer characteristic is received by the receiver 60 of FIG. 5 as the continuous time signal y (t). It The signal y (t) is sample y _n = y
Sampled by analog-to-digital converter 61 at a sampling rate f _s = 1 / T at a time determined by local clock 58 to produce (nT).
Sample y (nT) is processed by a timing adjuster 62 which generates the next corrected sample y _'n.

According to known sampling principles, the transmitted continuous-time PAM passband signal y (t) has the following formula:

[0038]

[Equation 12]

Can be reconstructed from Where y (t) is B
Bandlimited to Hz and f _s ≧ 2B. For example, B
Is 3000 Hz. In a real system it is not possible to use an infinite number of samples to reconstruct a continuous-time waveform, but rather just y (nT) 2N + 1
Values can be used, where N is a finite integer. In the time range of (n−N) T ≦ t ≦ (n + N) T, the waveform y (t) is expressed by

[0040]

[Equation 13]

Can be approximated by Where In '
(T) is the raised cosine function

[0042]

[Equation 14]

Where sinc (tf _s ) = sin (πt
f _s ) / πtf _s . The parameter a is known as the roll-off factor and is equal to 0.1 by way of example. The variable number v is within the variable equal to nT + τ in equation (8). Equation (8) is

[0044]

[Equation 15]

If np = i

[0046]

[Equation 16]

Can be rewritten as The variable number τ reflects the phase difference between the corrected sample y ′ = y (nT + τ) and the uncorrected received sample y (nT) obtained by the local clock. It is called the phase difference variable. It can be seen that the convolution of equation (9) equation (8) can be evaluated at a particular value of the phase difference variable to obtain samples y _'n corrected. When τ ₀ is the optimum phase difference and τ = τ ₀ , the corrected sample y ′ _n is the optimum sample.

The timing adjuster 62 of FIG. 5 has the equation (8)
Is a filter that evaluates the convolution of with a specific value of the variable v = nT + τ to obtain a corrected sample y ′ _n .
The value of the phase difference variable τ is determined by the timing difference detector 70. The timing difference detector 70 sets the value τ to the optimum value τ ₀
Adjust until you reach. The operation of the timing difference detector 70 will be described below.

As shown in FIG. 5, the timing adjuster 62
The corrected samples y ′ _n output from the are processed by the passband PAM receiver 64. Passband PA
The operation of M receiver 64 is shown in more detail in FIG. Sample y _'n is the impulse response function

[0050]

[Equation 17]

Or in the frequency domain

[0052]

[Equation 18]

Is processed by the filter 80 at. Since the impulse response is complex, two real filters with impulse responses Re {h (mT)} and Im {h (mT)} are needed for this implementation. The receive filter 80 preferably does not pass negative frequencies.
This is the output signal

[0054]

[Formula 19]

Means that does not include negative frequency components. By way of example, the filter f (mT) is a simple low pass filter with a gain of 1 in the pass band. The signal w (mT) is a coefficient for generating the output signal r (mT) = r _r (mT) + jr _i (mT) in the demodulator 82.

[0056]

[Equation 20]

It is multiplied by. It is clear that for a noise-free and distortion-free transmission channel, m = n and s (nT) = r (mT). As indicated above, the timing adjuster 62 of FIG. 5 uses the variable convolution of equation (8) with variable v = n.
It is a filter that evaluates with a specific value of T + τ. Symbol synchronization techniques are used to select the value τ of the phase difference variable.

FIG. 7 shows a waveform having a unipolar RZ format (in continuous time display). The power spectral density of this signal has a delta function at the symbol rate. As a result, a symbol clock signal is obtained by passing the unipolar RZ waveform through a narrow bandpass filter tuned to the symbol rate. However, in general, the output of the passband PAM receiver 64 of FIG.
Does not have Z format. In this case the passband PAM
It is necessary to convert the output of receiver 64 into a unipolar RZ waveform. For example, the output of passband PAM receiver 64 may have a polar NRZ format.
The polar NRZ format is shown in continuous time display in FIG. (The continuous time representation is used for clarity in FIG. 8. However, because the receiver 64 of FIG. 5 utilizes digital signal processing, the output of the receiver 64 is actually a series of samples at the sampling rate. A square circuit may be used to convert the output of the passband PAM receiver into RZ unipolar format. Therefore, in the receiver of FIG. 5, the squaring circuit 66 is

[0059]

[Equation 21]

Calculate The output of squaring circuit 66 is shown in FIG. 9 (in continuous time for clarity). A narrow bandpass filter 68 tuned to the symbol frequency then filters the output signal of the squaring circuit 66. Thus, the output of the narrow bandpass filter 70 is positive at the optimal sampling time, which is, for example, a sine wave with a frequency f _b = 1 / T _b equal to that of the symbol clock, ie, cos (2πf _b mT + φ). A symbol clock signal with peaks and symmetrical on each side. Such a signal (in continuous time representation for clarity) is shown in Figure 10 (Leon W. Coach).
II, "Digital-Analog Communication System", 3rd edition, 1
990, Macmillan Publishing Company, New York, p. 157).

The timing difference detector 70 of FIG. 5 receives the clock signal output from the narrow bandpass filter 68 and uses the value for τ used in the timing adjuster 62 to evaluate the convolution of equation (8). To decide. More specifically, the value τ of the phase difference variable is selected by the timing difference detector 70 for evaluating the convolution of equation 8 to obtain y ′ _n such that the corrected sample y ′ _n is closer to the optimal sample. It It should be noted that τ → constant when the corrected sample is the optimal sample. When this situation is met, it means that the local and remote clocks are in sync.

The corrected sample y ′ _n is the phase difference variable τ
Is called the optimal sample when has the optimal value τ ₀ . The time variable v = nT + τ ₀ used to evaluate the convolution of equation (8) to obtain the optimum sample is referred to as the optimum sampling time (for the passband signal) and τ ₀ is the optimum phase difference. Be told. The symbol sample r (mT) obtained at the optimum sampling time at the output of the PAM receiver 64 of FIG. 5 (for example, which is the symbol clock signal having a positive peak at 2400 Hz) is transmitted by the transmitter which symbol from the alphabet. Easily processed to identify what is done.

Since the sampling rate 1 / T is, for example, three times the symbol rate R = 1 / T _b , the periodic symbol clock signal output by the filter 68 of FIG. 5 during one period T _b . There are 3 samples. If the corrected sample is the optimal sample, the corresponding symbol clock signal is shown in FIG. 11A, which is shown in FIG.
It is an expanded portion of one period of the zero signal period T _b . If there is no optimal sample among the corrected samples, the corresponding symbol clock signal is shown in FIG. 11 (B) or FIG. 11 (C).

Therefore, FIG. 11A shows the symbol clock signal when the corrected sample is the optimum sample, and FIG. 11B shows when the corrected sample is delayed from the optimum sample. Figure 1 shows the symbol clock of
1 (C) shows the symbol clock when the corrected sample is ahead of the optimal sample. The operation of the timing difference detector 70 will now be considered.

The local receiver clock is said to be synchronous to the transmitter clock when it has the correct frequency and phase relationship with respect to the received input signal, due to the propagation delay through the channel. The transmitter is delayed when compared to the clock. In other words, the local receiver clock is synchronized with the input symbol stream at the transmitter.

If the local and remote clocks are synchronized, the symbol clock signal is

[0067]

[Equation 22]

Then, the result is as shown in FIG. In this case, the phase difference τ is substantially constant and the θ value is substantially zero. τ
The initial value of is obtained by the timing difference detector 70 from the value of θ that separates S1 and the peak of the cosine wave in FIG. 11B. Therefore, as can be seen from FIG. 11B, -θ / 120 = τ / T (11) S1 = Acosθ (12) S2 = Acos (θ + 120 °) (13) From Equations 12 and 13 A (amplitude of sine wave) By removing the following, the following equation is obtained for θ.

[0069]

[Equation 23]

Here, the initial value of τ is obtained from equation (11). After the initial value of the phase difference τ is calculated, the τ value is adjusted once during each symbol rate interval T _b = 3T.
This adjustment is performed until synchronization is achieved and τ → constant.
If S3> S2 in each period T _b , it means that the corrected sample lags behind the optimum sample, and τ is τ → τ−Δτ when Δτ is a constant discrete step size. Should be adapted as is. If S3 <S2, this means that the corrected sample is ahead of the optimal sample, and τ is τ →
It should be adapted to be τ + Δτ.

It should be noted that this technique of adjusting τ does not indicate whether the local receiver clock is very slow or very fast compared to the remote clock. However, in general, when the local clock is slower than the remote clock, S3> S2 is more likely than S3 <S2. If the local clock is faster than the remote clock, then S3 <S2 is more likely than S3> S2.

Asynchronous problems, as described above, result in one sample of the received signal being lost when the local clock is too slow, or an extra portion of the received signal when the local clock is too fast. A sample will be obtained. More particularly, if τ is less than a predetermined limit, such as for τ <−0.875, it means that the local receiver clock is slower than the remote transmitter clock and one corrected sample (ie, the next passivant PAM reception). Input to the vessel) is lost. To correct this, the phase difference variable used to evaluate equation (8) is first changed to τ → τ + 1.0. Then, before the next received sample is fed to the timing adjuster,
The new phase difference variable τ is used to evaluate the convolution of equation (8) to generate additional corrected samples.

If τ is greater than the predetermined limit, such as τ> 0.875, it means that the local receiver clock is faster than the remote transmitter clock and one extra corrected sample data is obtained. . To correct this, the next received sample is processed by the timing adjuster and the corrected sample generated by the convolution of equation (8) is discarded. The phase difference variable used to evaluate equation (8) is then changed to τ → τ-1.0.

For each value of τ produced by the timing difference detector 70, the 2N + 1 value of the raised cosine function is needed by the timing adjustment filter 62 to evaluate the convolution of equation (8). One approach to evaluating the raised cosine function is to store it in a memory table of raised cosine functions. However, the drawback of this approach is that it requires a large memory capacity. An alternative to this approach is to store a finite set of values of the raised cosine function in memory and then use the Taylor series to evaluate the raised cosine function at other values.

In the above description, the receiver according to the invention of FIG. 5 has been seen as consisting of a plurality of interconnected circuit elements. While this is one approach for implementing the receiver of the present invention, in another approach the receiver is under software control a single digital signal processor or microprocessor that performs the functions shown in FIG. It is carried out by using.

Finally, the above-described embodiments of the present invention are intended to be examples only. Many other embodiments will be apparent to those of ordinary skill in the art without departing from the spirit and scope of the appended claims.

[Brief description of drawings]

FIG. 1 is a schematic illustration of a baseband PAM system.

FIG. 2 is an exemplary diagram of a PAM passband transmitter.

3A and 3B are exemplary diagrams of symbol alphabets used for a passband PAM.

FIG. 4 is an exemplary diagram of another passband transmitter.

FIG. 5 is a diagram illustrating a PAM passband receiver according to an embodiment of the present invention.

FIG. 6 is a diagram illustrating a PAM passband receiver according to an embodiment of the present invention.

FIG. 7 is a diagram of signals utilized in the receiver shown in FIG.

FIG. 8 is a diagram of the signals utilized in the receiver shown in FIG.

9 is a diagram of the signals utilized in the receiver shown in FIG.

FIG. 10 is a diagram of signals utilized in the receiver shown in FIG.

11 (A), (B) and (C) are diagrams of signals utilized in the receiver shown in FIG.

[Explanation of symbols]

 58 local clock 59 channel 60 receiver 61 analog-digital converter 62 timing adjuster 64 passband PAM receiver 66 square circuit 68 narrow band pass filter 70 timing difference detector 80 filter 82 demodulator

 ─────────────────────────────────────────────────── ─── Continuation of the front page (72) Inventor Qi Shan Yu Taiwan Taipei Yun Ho Fung Ho E Road Section 22 22 (72) Inventor Yen Chun Lin Taiwan Tofeng Tung Shin Road 124

Claims (5)

[Claims] 1. An apparatus for receiving a continuous time pulse amplitude modulated passband signal, wherein said received continuous time pulse amplitude modulated signal is cycled at a sampling rate to generate received samples under the control of a local clock. Sampling means for dynamically sampling, timing adjustment means for evaluating the convolution of the received sample and the impulse response function with a specific value of the phase difference variable to obtain a corrected sample, and a narrow bandpass filter, Means for deriving from the corrected sample a signal element having a period equal to the symbol period of the pulse amplitude modulated signal, and for each symbol period determining whether the corrected sample is an early or late delayed optimal sample To compare the series of samples of the signal element output from the deriving means to obtain the corrected sample Timing difference detector that varies the phase difference variable used to estimate the convolution when the leading and trailing optimum samples are varied by a constant discrete step size in each symbol time period. An apparatus for receiving a continuous time pulse amplitude modulated passband signal, comprising: means. 2. A passband pulse amplitude modulation receiver connected to the output of the timing adjustment means, and means for processing the signal output by the interpassband pulse amplitude modulation receiver to convert it to a unitary polar format. An apparatus having a narrow band at a symbol frequency for filtering the signal in a unitary polar format to derive the signal element whose period is the symbol period. 3. An apparatus for receiving a pulse amplitude modulated signal, the sampling means sampling the pulse amplitude modulated signal at a sampling rate that produces the received sample, and convolution of the received sample and the impulse response function. From the corrected sample to the symbol frequency of the pulse-amplitude-modulated signal, and a filter having a narrow transmission band for evaluating with a specific value of the phase difference variable to obtain a corrected sample. And a means for deriving a periodic signal at, and once the variable phase difference change used to evaluate the convolution during each period of the periodic signal is very large or very small. The phase difference which is determined and is variable according to a certain discrete step size is set to be very large or very small. If have a device for receiving the more becomes the pulse amplitude modulation signal timing difference detector means for varying. 4. The timing difference detection means compares a series of samples of the periodic signal in each period thereof to determine whether the corrected sample is a leading or trailing optimal sample and determines the discrete A step size that varies the phase difference variable used to evaluate the convolution if the corrected sample is the optimum sample that is ahead or behind in each symbol time period. The device according to. 5. A step of periodically sampling the pulse amplitude modulated signal to generate received samples, and a variable phase difference to obtain convolution-corrected samples of the received samples and the impulse response function. And electronically deriving a periodic signal having a period equal to the symbol period of the pulse amplitude modulated signal from the corrected sample using a narrow transmission band filter, And electronically comparing a series of samples to determine whether the corrected signal is the leading or trailing optimal sample in each period of the periodic signal, and the corrected sample Is an optimal sample that is either early or late, the phase difference variable used to evaluate the convolution is fixed by a constant discrete step size. A method of receiving a pulse amplitude modulation signal, which comprises the steps of: