Signal Delay Reconstruction Method Based on Dynamic Index and Complex-Coefficient Lagrange Interpolation*

This paper presents a large-range, high-precision and continuously variable delay reconstruction method for wideband and arbitrary bandlimited signal, which combines dynamic index technique with complex-coefficient Lagrange interpolation technique. The method samples time-continuous bandlimited signal and stores samples in sequence. It manages to obtain the high-precision delay parameters of every sampling period from desired delay to compute the so-called index position variable and interpolator parameters. It dynamically in-dexes and chooses a set of samples to implement piecewise complex-coefficient Lagrange interpolation for reconstructing the delayed sequences. The time-continuous delay reconstruction signal can be simply accomplished through digital-to-analog conversion. The mathematical model of the method and its transformed form is given, and the arithmetic of dynamic index and complex-coefficient Lagrange interpolation is derived. Simulation and test results show the validity and performance of the method.

SECTION I. Introduction

In signal processing field, delay reconstruction of ban-dlimited signal is often desired. It has found applications in many fields such as phased-array antenna processing^[1], time delay estimation ^[2], time delay calibration in multi-channel systems, transmission delay simulation in signal and channel simulator ^[3], and so on. For some applications, a constant delay or small-range and slowly variable delay may be sufficient and numerous analog or digital means can be employed. However, in some applications such as transmission delay simulation, the large-range, high-precision and continuously variable delay must be needed. In this case, the process is more difficult and it is significant to develop the large-range, high-precision and continuously variable signal delay reconstruction method.

For delaying a signal waveform by a given time, the original method is based on analog delay lines^{[4], [5]}. However, it is gradually substituted by digital means due to the low control-lable and low flexibility of analog means. The simplest digital delay reconstruction method is to sample the bandlimited signal and simply store the samples in a buffer memory for a given time. The delay range lies on the capabilities of buffer memory and delay precision is severely restricted by the sampling interval. If variable delay is desired, the write interval and read interval of buffer memory must be designed to be incommensurate. As a result, this method is only well suitable for the constant delay with lower precision. For delaying a signal wave by nonintegral multiplies of sampling inter-val, fractional delay filter works perfectly^{[6], [7]}. In this case, the delayed signal is reconstructed by means of bandlimited interpolation and an effective implementation structure^[8] can be employed for rapid update of fractional delay. However, when large-range delay is needed, the fractional delay filter must have adequately high orders, which increases computational cost and filter complexity, and even cannot be ac-complished when the desired delay is much larger than the sampling interval. Authors have proposed a delay reconstruction method based on equal-interval sampling and unequal-interval reconstruction ^[3]. It samples the bandlimited signal by an equal-interval sampling clock and outputs sampling sequences by a delayed unequal- interval clock. Its basic concept is to change the delay problem of wideband and unknown signal into the delay problem of known equal-interval sampling clock, because the high-precision and continuously variable delay of known sampling clock is more easy to achieve by high-order Direct digital synthesizer (DDS). This method can achieve a large-range, high-precision and continuously variable delay. However, it is considerably complex and hard to control due to the employment of analog unequal-interval clock.

In this paper, we present a signal delay reconstruction method based on dynamic index and complex-coefficient La-grange interpolation that can reconstruct large-range and high-precision and continuously variable delay signal for wide- band and arbitrary bandlimited signal. Next section derives the proposed method and gives the explicit reconstruction for-mula. Then the transformed form of the method is derived in Section III. Section IV gives the performance and the conclusion follows in Section V.

SECTION II. Signal Delay Reconstruction Based on Dynamic Index and Complex-Coefficient Lagrange Interpolation

The signal delay reconstruction method based on dynamic index and complex-coefficient Lagrange interpolation is derived, and then the explicit reconstruction formula is given. Al-though we assume that the original bandlimited signal is time-continuous and our objective is to reconstruct its time-continuous delayed signal, the proposed method is entirely suitable for the delay reconstruction of digital signal.

The signal delay reconstruction problem can be modeled as

$$\begin{equation*} S_{\tau}(t)=S[t-\tau(t)]\tag{1}\end{equation*}$$ where $S(t)$ is the original signal and assumed to be wideband, arbitrary and bandlimited, $\tau(t)$ is the desired delay. Our objective is to reconstruct the continuously variable delay signal $S_{\tau}(t)$.

We sample time-continuous bandlimited signal $S(t)$ at a constant interval $t=nT_{s}$ to obtain its samples $\{S(nT_{s})\}$, and then $S(t)$ can be expressed by its samples

$$\begin{equation*} S(t)=\sum_{n=-\infty}^{+\infty}S(nT_{s})h_{I}(t-nT_{s})\tag{2}\end{equation*}$$ where $h_{I}(t)$ is the impulse response of ideal time-continuous interpolating filter.

Thus Eq.(1) can be rewritten as

$$\begin{align*} S_{\tau}(t) & =S[t-\tau(t)] \\ & =\sum_{n=-\infty}^{+\infty} S(n T_{s}) h_{I}[t-\tau(t)-n T_{s}]\tag{3}\end{align*}$$

The entirely digital form can be obtained by introducing a fictitious analog-to-digital conversion operation in which the sampling interval is $t=kT_{s}$, and then Eq.(3) becomes

$$\begin{align*} S_{\tau}(k T_{s}) & =S[k T_{s}-\tau(k T_{s})] \\ & =\sum_{n=-\infty}^{+\infty} S(n T_{s}) h_{I}[k T_{s}-\tau(k T_{s})-n T_{s}]\tag{4}\end{align*}$$ where $n$ is defined as the discrete time variable of original signal and $k$ is the discrete time variable of reconstruction signal.

The form of Eq.(4) suggests that the digital delay reconstruction signal $S_{\tau}(kT_{s})$ can be reconstructed from samples $\{S(nT_{s})\}$ and digital impulse response $h_{I}[kT_{s}-\tau(kT_{s})]$. If a proper sampling interval is identified, the time-continuous delay reconstruction signal can be easily obtained by digital-to-analog conversion. Unfortunately, Eq.(4) is only conceptually feasible for large-range delay because the orders of interpolating filter will increase with the desired delay. As a result, we define new variables and substitute the original variables to obtain a more useful form.

Define variable $m_{k}$ which is named as index position variable

$$\begin{align*} m_{k} & =\text{int}\left\lceil\frac{k T_{s}-\tau(k T_{s})}{T_{s}}\right\rceil \\ & =\text{int}\left\lceil\frac{k T_{s}-p_{i n}(k T_{s}) T_{s}-p_{d}(k T_{s}) T_{s}}{T_{s}}\right\rceil \\ & =k-p_{i n}(k T_{s})\tag{5}\end{align*}$$ where int $\lceil z\rceil$ means the smallest integer exceeding $z. p_{in}(kT_{s})$ is the integer-interval numbers of $\tau(kT_{s})$ and $p_{d}(kT_{s})$ is the fraction-interval numbers, that is $$\begin{equation*}\begin{cases} p_{in}(kT_{s})= \text{int} \left\lfloor\frac{\tau(kT_{s})}{T_{s}}\right\rfloor\\ p_{d}(kT_{s})=\frac{\tau(kT_{s})}{T_{s}}-p_{in}(kT_{s})\end{cases}\tag{6}\end{equation*}$$ where int $\lfloor z\rfloor$ means the largest integer not exceeding $z. p_{in}(kT_{s})$ is an integer, $p_{d}(kT_{s})$ is an decimal and $p_{d}(kT_{s})\in [0,1)$.

Also, define the fractional interval

$$\begin{align*}\mu_{k} & =m_{k}-\frac{k T_{s}-\tau(k T_{s})}{T_{s}} \\ & =p_{d}(k T_{s})\tag{7}\end{align*}$$ and the interpolator variable $$\begin{equation*} i=m_{k}-n\tag{8}\end{equation*}$$

Substitute Eqs.(5)-(8) into Eq.(4) to show that it can be rewritten as

$$\begin{align*} S_{\tau}(k T_{s}) & =S[k T_{s}-\tau(k T_{s})]=S[(m_{k}-\mu_{k}) T_{s}] \\ & =\sum_{i=I_{1}}^{I_{2}} S[(m_{k}-i) T_{s}] h_{I}[(i-\mu_{k}) T_{s}]\tag{9}\end{align*}$$

Variables in Eq.(9) are: interpolator variable, index position variable and fractional interval. If the interpolating filter has Finite impulse response (FIR), the range $[I_{1}, I_{2}]$ of interpo-lator variable $i$ is fixed and finite. The index position variable $m_{k}$, determined by the integer-interval part of desired delay, indicates a position of signal samples $\{S(nT_{s})\}$. Around this position, $I=I_{2}-I_{1}+1$ signal samples should be chosen for the k-th computation. The fractional interval $\mu_{k}$ is used to identify $I$ digital impulse response to be employed for k-th computation. Comparing with Eq.(4), the integer-interval part of desired delay has been used to identify the index position variable, and the digital impulse response $h_{I}[(i-\mu_{k})T_{s}]$ is only determined by the fraction-interval part of the desired delay. Thus, the digital delay reconstruction signal can be easily obtained by dynamically choosing a set of signal samples to implement a piecewise fraction-interval interpolation.

There are numerous functions that interpolating filter could be based upon^{[9], [10]}. In this paper, we introduce La-grange interpolation to give an explicit reconstruction formula for $S_{\tau}(kT_{s})$. The Lagrange interpolation coefficients can be given as

$$\begin{equation*} C_{i}(t)=\prod_{j=I_{1},j\neq i}^{I_{2}}\frac{t-t_{j}}{t_{i}-t_{j}}\tag{10}\end{equation*}$$

Then digital impulse response of interpolating filter can be computed as

$$\begin{equation*} h_{I}^{0}[(i-\mu_{k})T_{s}]=C_{i}(\mu_{k}T_{s})=\prod_{j=I_{1},j\neq i}^{I_{2}}\frac{\mu_{k}-j}{i-j}\tag{11}\end{equation*}$$

Refs. [11], [12] have proved that interpolating filter with the digital impulse response of Eq.(11) is a good approximation of fractional delay system with the ideal frequency response of $\mathrm{e}^{-\mathrm{j}\omega\mu_{k}}$. The superscript ‘0’ denotes that it has a maximally flat frequency characteristic just around $\omega_{0}=0$ where $\omega_{0}$ is the normalized angular frequency. To make the maximally flat region around an arbitrary angular frequency, a complex-coefficient form can be obtained by modulating thus:

$$\begin{align*} h_{I}[(i-\mu_{k}) T_{s}] & =\mathrm{e}^{\mathrm{j} \omega_{0}(i-\mu_{k})} h_{I}^{0}[(i-\mu_{k}) T_{s}] \\ & =\mathrm{e}^{\mathrm{j} \omega_{0}(i-\mu_{k})} \prod_{j=I_{1}, j \neq i}^{I_{2}} \frac{\mu_{k}-j}{i-j}\tag{12}\end{align*}$$

Both coefficients in Eqs.(11) and (12) are dependent on the fractional interval, which is disadvantageous for continuous delay control. So let each coefficient be approximated by L-order polynomial of fractional interval, that is

$$\begin{equation*} h_{I}^{0}[(i-\mu_{k})T_{s}]=\sum_{l=0}^{L}c_{l}(i)\mu_{k}^{l}\tag{13}\end{equation*}$$ where $c_{l}(i)$ are fixed and real-valued approximating coefficients. Then Eq.(12) becomes $$\begin{align*} h_{I}[(i-\mu_{k}) T_{s}] & =\sum_{l=0}^{L} c_{l}(i) \mu_{k}^{l} \mathrm{e}^{\mathrm{j} \omega_{0}(i-\mu_{k})} \\ & =\sum_{l=0}^{L} d_{l}(i) \mu_{k}^{l} \mathrm{e}^{-\mathrm{j} \omega_{0} \mu_{k}}\tag{14}\end{align*}$$ where $d_{l}(i)=c_{l}(i)\mathrm{e}^{-\mathrm{j}\omega_{0}i}$ are fixed and complex-value coefficients.

Thus, the explicit reconstruction formula can be obtained as

$$\begin{align*} S_{\tau}(k) & =\sum_{i=I_{1}}^{I_{2}} S(m_{k}-i) \sum_{l=0}^{L} d_{l}(i) \mu_{k}^{l} \mathrm{e}^{-\mathrm{j} \omega_{0} \mu_{k}} \\ & =\sum_{l=0}^{L}\left[\sum_{i=I_{1}}^{I_{2}} d_{l}(i) S(m_{k}-i)\right] \mu_{k}^{l} \mathrm{e}^{-\mathrm{j} \omega_{0} \mu_{k}}\tag{15}\end{align*}$$

The form of Eq.(15) gives us an effective implementation structure to reconstruct $S_{\tau}(k)$ in real time. According to the index position variable $m_{k}$, we choose $I$ samples from $\{S(nT_{s})\}$ to pass through a set of parallel filter with fixed and complex-valued coefficients $d_{l}(i)$, and then the outputs are summed after each of them is weighted by $\mu_{k}^{l}\mathrm{e}^{-\mathrm{j}\omega_{0}\mu_{k}}$. If the frequency of signal $S(t)$ is far less than the sampling rate, the explicit reconstruction formula can be simplified by using real-valued coefficients $c_{l}(i)$, and making weighting coefficient $\mu_{k}^{l}\mathrm{e}^{-\mathrm{j}\omega_{0}\mu _{k}}$ become $\mu_{k}^{l}$. Thus, if a proper sampling interval is identified, the time-continuous delay reconstruction signal $S_{\tau}(t)$ can be simply achieved through digital-to-analog conversion.

SECTION III. Transformed Form

When deriving the signal delay reconstruction method in Section II, the frequency of original input signal is not re-stricted. However, when original signal has a higher frequency such as for a Radio frequency (RF) signal, high-speed devices must be applied and high-speed digital signal processing must be deal with. As a result, we present a transformed form of the proposed method.

The original signal $S(t)$ is assumed to be a bandlimited RF signal. Now, we change its form as

$$\begin{align*} S(t) & =S(t) \times \exp (-\mathrm{j} 2 \pi f_{R} t) \times \exp (\mathrm{j} 2 \pi f_{R} t) \\ & =S^{\prime}(t) \exp (\mathrm{j} 2 \pi f_{R} t)\tag{16}\end{align*}$$ where $f_{R}$ is a known reference frequency. If a proper value of $f_{R}$ is identified, signal $S^{\prime}(t)$ will be a bandlimited baseband or low-Intermediate frequency (IF) signal.

Then Eq.(1) can be rewritten as

$$\begin{align*} S_{\tau}(t) & =S[t-\tau(t)] \\ & =S^{\prime}[t-\tau(t)] \exp \{\mathrm{j} 2 \pi f_{R}[t-\tau(t)]\} \\ & =S^{\prime}[t-\tau(t)] \exp [-\mathrm{j} 2 \pi f_{R} \tau(t)] \exp (\mathrm{j} 2 \pi f_{R} t) \\ & =S_{\tau}^{\prime}(t) \exp (\mathrm{j} 2 \pi f_{R} t)\tag{17}\end{align*}$$ where $$\begin{equation*} S_{\tau}^{\prime}(t)=S^{\prime}[t-\tau(t)] \exp [-\mathrm{j} 2 \pi f_{R} \tau(t)]\tag{18}\end{equation*}$$

The form of Eq. (17) suggests that the delay reconstruction of RF signal can be implemented by first generating the baseband or low-IF reconstruction signal $S_{\tau}^{\prime}(t)$, and then mul-tiplying by a known complex exponential signal. In this case, Eq.(15) cannot be employed directly to generate the digital version of signal $S_{\tau}^{\prime}(t)$ and a new explicit reconstruction for-mula will be derived as follows.

Signal $S_{\tau}^{\prime}(t)$ is discretizated at a constant interval $t=kT_{s}$, and then we have

$$\begin{equation*} S_{\tau}^{\prime}(k T_{s})=S^{\prime}[k T_{s}-\tau(k T_{s})] \exp [-\mathrm{j} 2 \pi f_{R} \tau(k T_{s})]\tag{19}\end{equation*}$$ where $S^{\prime}[kT_{s}-\tau(kT_{s})$ can be computed as $$\begin{equation*} S^{\prime}[k T_{s}-\tau(k T_{s})]=\sum_{l=0}^{L}\left[\sum_{i=I_{1}}^{I_{2}} d_{l}(i) S^{\prime}(m_{k}-i)\right] \mu_{k}^{l} \mathrm{e}z^{-\mathrm{j} \omega_{0} \mu_{k}}\tag{20}\end{equation*}$$

Define a time-varying modified coefficient

$$\begin{align*} g(k T_{s}) & =\exp [-\mathrm{j} 2 \pi f_{R} \tau(k T_{s})] \\ & =\exp (-\mathrm{j} \omega_{R} p_{k})\tag{21}\end{align*}$$ where $p_{k}=\tau(kT_{s})/T_{s}$ and $\omega_{R}=2\pi f_{R}T_{s}$ is a known digital angular frequency.

Then $S_{\tau}^{\prime}(kT_{s})$ can be reconstructed as

$$\begin{align*} S_{\tau}^{\prime}(k T_{s}) & =S^{\prime}[k T_{s}-\tau(k T_{s})] \times g(k T_{s}) \\ & =\sum_{l=0}^{L}\left[\sum_{i=I_{1}}^{I_{2}} d_{l}(i) S^{\prime}(m_{k}-i)\right] \mu_{k}^{l} \mathrm{e}^{-\mathrm{j}(\omega_{0} \mu_{k}+\omega_{R} p_{k})}\tag{22}\end{align*}$$

Eq.(22) also gives us an effective implementation structure, in which only four parameters are needed: the baseband or low-IF input samples $\{S^{\prime}(nT_{s})\}$, index position variable $m_{k}$, fractional interval $\mu_{k}$ and the desired digital delay $p_{k}$. Thus, the delay reconstruction of RF signal can be simply implemented by employing Eq.(22) for a baseband or low-IF signal reconstruction and carrying through relevant frequency con-version.

SECTION IV. Performance

In this section, we will illustrate the performance of the method according to the explicit reconstruction formula as Eq. (l5).

For an arbitrary time $k$, the signal samples around a past time $n=m_{k}=k-p_{in}(k)$ should be chosen for the k-th computation, which means that at least $p_{in}(k)$ samples need to be stored in memory for an integer-interval delay. The numbers of stored samples are determined by integer-interval part of the desired delay and will dynamically change with it. Hence, when a large-range delay is needed, the proposed method can be achieved by simply expanding the capacities of memory. On the other hand, the desired delay (or its surrogate $m_{k}$ and $\mu_{k}$) can be updated in every sampling period, because coefficients $d_{l}(i)$ are fixed and entirely independent on the desired delay. Such characteristic makes proposed method be well suited for continuously variable delay control. The delay precision of proposed method mainly lies on the accuracy of sampling interval, the calculation errors of delay parameters, the quantization errors, the performance of digital interpolating filter, and so on. In this paper, the first three aspects are not concerned and more emphasis will be on the performance of interpolating filter.

The frequency response $H_{I}(\omega,\mu)$ of digital interpolating filter is computed, and then the performance will be analyzed by defining magnitude error $\varepsilon_{m}(\omega,\mu)$ and group delay error $\varepsilon_{\tau}(\omega,\mu)$ as

$$\begin{align*}\varepsilon_{m}(\omega, \mu) & =20 \log _{10} \frac{1}{|H_{I}(\omega, \mu)\vert}, \omega \in[0.2 \pi, 0.8 \pi], \mu \in[0,1)\tag{23} \\ \varepsilon_{\tau}(\omega, \mu) & =\vert\mu-\tau(\omega, \mu)\vert, \omega \in[0.2 \pi, 0.8 \pi], \mu \in[0,1)\tag{24}\end{align*}$$ where $\tau(\omega,\mu)$ is the group delay response $$\begin{equation*}\tau(\omega,\mu)=-\frac{\mathrm{d}\{\arg[H_{I}(\omega,\mu)]\}}{\mathrm{d}\omega}\tag{25}\end{equation*}$$ and $\arg[H_{I}(\omega,\mu)]$ is the phase response of interpolating filter.

We suppose that normalized angular frequency range of original signal is $\omega\in[0.2\pi, 0.8\pi]$, the frequency response of complex-coefficient Lagrange interpolator is designed to have maximally flat characteristic around $\omega_{0}=0.5\pi$. The order of Lagrange interpolation is assumed to be 7 and each of coefficients is approximated by a 7-order polynomial. Fig. 1 and Fig. 2 respectively illustrate the magnitude error and group delay error versus fractional interval and normalized angular frequency. For $\omega\in[0.2\pi, 0.8\pi]$ and $\mu\in [0,1)$, the maximum magnitude error is only $4.694 \times 10^{-3}\text{dB}$ and the maximum group delay error will not exceed 7.376 $\times 10^{-4}$ samples.

SECTION V. Conclusion

The signal delay reconstruction method based on dynamic index and complex-coefficient Lagrange interpolation is proposed. In Section II, the method is derived in detail and the explicit reconstruction formula is given. Then we present a transformed form of the proposed method for RF signal delay reconstruction. In Section IV, the performance is analyzed and the results show its validity and good performance.

Fig. 1.

Magnitude error versus fractional interval and normal-ized angular frequency

Fig. 2.

Group delay error versus fractional interval and normalized angular frequency