### The LCT domain analysis of decimation and interpolation

To study the decimation and interpolation in the LCT domain, in this section, we firstly give the definition of equivalent filter in the LCT domain. Then, the direct implementation structure for decimation and interpolation in the LCT domain are derived based on the definition. Moreover, the polyphase networks for decimation and interpolation in the LCT domain are also deduced.

#### Equivalent FIR filter in the LCT domain

The convolution theorem for signals in the LCT domain have been introduced in Wei et al. (2012), Deng et al. (2006). Analogously, based on the definition of DTLCT, the convolution theorem in the DTLCT domain can be expressed as:

$$\begin{aligned} f(nT)\,=\, & {} \sqrt{\frac{1}{{j2\pi b}}} e^{ - j\frac{a}{{2b}}(nT)^2 } \left[ \left( h(nT)e^{j\frac{a}{{2b}}(nT)^2 } \right) *\left( x(nT)e^{j\frac{a}{{2b}}(nT)^2 }\right) \right] \end{aligned}$$

(8)

$$\begin{aligned} F_A (w)\,=\, & {} e^{ - j\frac{{dw^2 }}{{2bT^2 }}} X_A (w)H_A (w) \end{aligned}$$

(9)

where \(F_A (w)\), \(X_A (w)\), \(H_A (w)\) denotes the DTLCT of signal *f*(*t*), *x*(*t*), *h*(*t*), respectively. From Eq. (9), it is easy to know that the convolution theorem for the DTLCT contains an extra chirp factor and hence does not easily implement in the time domain. On the other hand, by multiplying the \(e^{ - j\frac{{dw^2 }}{{2bT^2}}}\) to both sides of Eq. (9), we can obtain

$$\begin{aligned} F_A (w)e^{ - j\frac{{dw^2 }}{{2bT^2 }}} = X_A (w)e^{ - j\frac{{dw^2 }}{{2bT^2 }}} H_A (w)e^{ - j\frac{{dw^2 }}{{2bT^2 }}} \end{aligned}$$

(10)

According the definition of simplified LCT, we get

$$\begin{aligned} \bar{Y}_A (w) = \bar{X}_A (w)\bar{H}_A (w) \end{aligned}$$

(11)

Equation (11) shows that the convolution of two signals is equivalent to simple multiplication of their simplified LCTs in the simplified LCT domain. It is more useful in practical filtering. Based on this, we give the definition of the equivalent filter in the LCT domain as follows.

###
**Definition 1**

Suppose \(H_A (w)\) denotes the DTLCT of the finite length sequence *h*(*nT*), then we define \(\bar{H}_A (w)\) as the equivalent filter in the LCT domain.

$$\begin{aligned} \bar{H}_A (w) = H_A (w)e^{ - j\frac{{dw^2 }}{{2bT^2 }}} \end{aligned}$$

(12)

This definition shows that the equivalent FIR filter in the LCT domain does not contain an extra chirp factor and is easy to implement in the time domain. In addition, when substituting special parameters into the equivalent FIR filter for the LCT, the equivalent FIR filter in the FRFT domain can be obtained.

#### The polyphase implementation of decimation and interpolation in the LCT domain

The polyphase decomposition is very fundamental to the efficient implementation of decimation and interpolation in the multirate digital signal processing systems. It can be applied for the derivation of new sampling theorems and the recovering bandlimited signal from nonuniformly sampled versions. Theories and applications of polyphase decomposition for the decimation and interpolation in the FT and FRFT domain have been well studied in Vaidyanathan (1990), Tao et al. (2008), Meng et al. (2007). To obtain the polyphase implementation of decimation and interpolation in the LCT domain, we study the direct implementation structure for decimation and interpolation in the LCT domain at first.

Now, let us consider the definition of equivalent FIR filter, the block diagram notation of decimation process in the LCT domain can be depicted in Fig. 1.

Then, according to the convolution theorem in the LCT domain, the direct implementation structure for decimation process in the LCT domain can be obtained in Fig. 2.

From Fig. 2, it is easy to show that the direct implementation structure for decimation in the LCT domain is inefficient due to every calculation \(y(n_1 T_1 )\) needs to be completed within a \(T_1\). However, the calculation efficiency can be improved by utilizing the equivalent FIR filter and exchanging the \(e^{ - j\frac{{dw^2 }}{{2bT^2 }}}\) and the decimator. Thus, the direct efficiency implementation structure for decimator in LCT domain can be obtained in Fig. 3. It shows that the calculated amount in Fig. 3 reduced to 1 / *D* of the direct implementation structure in Fig. 2. Similar to the decimator case, the efficiency implementation structure for interpolator in the LCT domain also can be obtained.

From above analysis, the direct efficiency implementation structure for decimator and interpolator in the LCT domain have been derived. To obtain the polyphase networks for decimation and implementation in the LCT domain, the polyphase networks for equivalent filter have been derived in the following at first.

Let \(\bar{H}_A (w) = H_A (w)e^{ - j\frac{{dw^2 }}{{2bT^2 }}}\) be an equivalent filter, *H*(*w*) is written in the form

$$\begin{aligned} H(w) =&\sum \limits _{n = 0}^{Q - 1} {h(nD + 0)e^{j\frac{{dw^2 }}{{2bT^2 }}} e^{j\frac{a}{{2b}}(nD)^2 T^2 } e^{ - j(nD)w/b} } \\&+ \sum \limits _{n = 0}^{Q - 1} {h(nD + 1)e^{j\frac{{dw^2 }}{{2bT^2 }}} e^{j\frac{a}{{2b}}(nD + 1)^2 T^2 } e^{ - j(nD + 1)w/b} } \\&+ \cdots \\&+ \sum \limits _{n = 0}^{Q - 1} {h(nD + k)e^{j\frac{{dw^2 }}{{2bT^2 }}} e^{j\frac{a}{{2b}}(nD + k)^2 T^2 } e^{ - j(nD + k)w/b}} \end{aligned}$$

(13)

where \(N = QD\) stands for the length of FIR filter in the LCT domain, and

$$\begin{aligned} &\sum \limits _{n = 0}^{Q - 1} {h(nD + k)e^{j\frac{{dw^2 }}{{2bT^2 }}} e^{j\frac{a}{{2b}}(nD + k)^2 T^2 } e^{ - j(nD + k)w/b} }\\&\quad = e^{ - jkw/b} e^{j\frac{a}{{2b}}k^2 T^2 } e^{j\frac{{dw^2 }}{{2bT^2 }}} \sum \limits _{n = 0}^{Q - 1} {h(nD + k) } \times e^{j\frac{a}{{2b}}(n^2 D^2 + 2knD)T^2 } e^{ - jnDw/b} \end{aligned}$$

(14)

Then, suppose

$$\begin{aligned} E_k (Dw)&= e^{j\frac{{dw^2 }}{{2bT^2 }}} \sum \limits _{n = 0}^{Q - 1} {\left( h(nDT + kT)e^{j\frac{{2aknDT{}^2}}{{2b}}} \right) } \times e^{j\frac{{an^2 D^2 T{}^2}}{{2b}}} e^{ - jnDw/b} \\&= F_A [g_k (nDT)] \end{aligned}$$

(15)

where

$$\begin{aligned} g_k (nDT) = h(nDT + kT)e^{j\frac{{2aknDT{}^2}}{{2b}}} \end{aligned}$$

(16)

Therefore, we can obtain

$$\begin{aligned} H(w) = \sum \limits _{k = 0}^{D - 1} {e^{ - jkw/b} e^{j\frac{a}{{2b}}k^2 T^2 } } E_k (Dw) \end{aligned}$$

(17)

According to Eq. (12), we can rewrite Eq. (17) as

$$\begin{aligned} \bar{H}(w) = \sum \limits _{k = 0}^{D - 1} {e^{ - jkw/b} e^{j\frac{a}{{2b}}k^2 T^2 } } \bar{E}_k (Dw) \end{aligned}$$

(18)

where

$$\begin{aligned} \bar{E}_k (Dw) = \sum \limits _{n = 0}^{Q - 1} {\left( h(nDT + kT)e^{j\frac{{2aknDT{}^2}}{{2b}}} \right) e^{j\frac{{an^2 D^2 T{}^2}}{{2b}}} e^{ - jnDw/b} } \end{aligned}$$

(19)

On the other hand, from the properties of the LCT, we get

$$\begin{aligned} F_A [\delta (nT - kT)]&= K_A e^{j\frac{{dw^2 }}{{2bT^2 }}} \sum \limits _{n = - \infty }^\infty {\delta (nT - kT)e^{j\frac{a}{{2b}}(nT)^2 } e^{ - jnw/b} } \\&= K_A e^{j\frac{{dw^2 }}{{2bT^2 }}} e^{j\frac{a}{{2b}}(kT)^2 } e^{ - jkw/b} \\ \end{aligned}$$

(20)

From above all, we obtain

$$\begin{aligned} H(w)&= \sum \limits _{k = 0}^{D - 1} {e^{ - jkw/b} e^{j\frac{a}{{2b}}k^2 T^2 } } E_k (Dw) \\&= \frac{1}{{K_A }}e^{ - j\frac{{dw^2 }}{{2bT^2 }}} \sum \limits _{k = 0}^{D - 1} {F_A [\delta (nT - kT)]} E_k (Dw) \\ \end{aligned}$$

(21)

Assume that

$$\begin{aligned} Y_A (w) = X_A (w)\bar{H}_A (w) \end{aligned}$$

(22)

Substituting Eq. (21) into Eq. (22) and rearranging it, we get

$$\begin{aligned} Y_A (w)e^{j\frac{{dw^2 }}{{2bT^2 }}}&= X_A (w)H_A (w) \\&= \frac{1}{{K_A }}\sum \limits _{k = 0}^{D - 1} {e^{ - j\frac{{dw^2 }}{{2bT^2 }}} } X_A (w)F_A [\delta (nT - kT)]E_k (Dw) \\ \end{aligned}$$

(23)

From the convolution theorem in the LCT domain, \(X_A (w)\) can be written as

$$\begin{aligned} X_A (w)&= \frac{1}{{K_A }}e^{ - j\frac{{dw^2 }}{{2bT^2 }}} X_A (w)F_A [\delta (nT - kT)] \\&= F_A \left[ e^{ - j\frac{{an^2 T^2 }}{{2b}}} x(nT - kT)e^{j\frac{{a(n - k)^2 T^2 }}{{2b}}} e^{j\frac{{ak^2 T^2 }}{{2b}}} \right] \\ \end{aligned}$$

(24)

From Eqs. (23) and (24), we have

$$\begin{aligned} Y_A (w)&= e^{ - j\frac{{dw^2 }}{{2bT^2 }}} E_k (Dw) \sum \limits _{k = 0}^{D - 1} {F_A \left[ e^{ - j\frac{{an^2 T^2 }}{{2b}}} x(nT - kT) e^{j\frac{{a(n - k)^2 T^2 }}{{2b}}} e^{j\frac{{ak^2 T^2 }}{{2b}}}\right] } \\&= \sum \limits _{k = 0}^{D - 1} {F_A \left[ e^{ - j\frac{{an^2 T^2 }}{{2b}}} x(nT - kT)e^{j\frac{{a(n - k)^2 T^2 }}{{2b}}} e^{j\frac{{ak^2 T^2 }}{{2b}}}\right] } \bar{E}_k (Dw) \end{aligned}$$

(25)

Based on this equation, the polyphase implementation of equivalent FIR filter in the LCT domain can be derived in Fig. 4.

Similarly, the polyphase implementation of \(\bar{E}_k (Dw)\) can be obtained in Fig. 5.

Then, according to Figs. 1 and 4, the polyphase implementation structure for decimation in LCT domain can be obtained in Fig. 6. Likewise, the polyphase implementation structure for interpolation in LCT domain also can be derived.

Furthermore, the polyphase implementation structure for decimation and interpolation in LCT domain have many real applications, such as it can perform digital filtering the general case, it also can offer significant saving in computation rate and hardware in two important applications, sampling rate alternation and realization of filter banks.