The biorthogonal Fourier transform (BFT) and its inverse conversion algorithm is defined in [29, 30] (or refer to [28]):

$$ \begin{aligned} \text{BFT}[f(t)]\triangleq F(\beta) & =2{\int\nolimits}_{0}^{+\infty}f(t)t\exp\left(-j\beta t^{2}\right)dt \\ \text{IBFT}[\!F(\beta)]\triangleq f(t) & =\frac{1}{2\pi}{\int\nolimits}_{-\infty}^{+\infty}F(\beta)\exp\left(j\beta t^{2}\right)d\beta \end{aligned} $$

(2)

Obviously, to the signal of linear frequency modulation *f*(*t*)=*e**x**p*(*j**π**K*_{
m
}*t*^{2}), it has BFT[*f*(*t*)]=2*π**δ*(*β*−*π**K*_{
m
}), where the impulse position at *π**K*_{
m
} in the BFT domain reflects the chirp rate. Referring to the short-term discrete Fourier transform (STDFT), the discrete BFT (D-BFT) performs a circular convolution operation. Here, the oversampling frequency is denoted as *f*_{
c
} and its interval is *T*_{
c
}. The time and BFT domain are discretized as *t*=*l**T*_{
c
} and *β*=*k**Δ**β*, where *l* and *k* are integers. Furthermore, the *N*-point symbol duration is defined as *T*_{
s
}=*N**T*_{
c
} with *N*≫1, and *N*-point chirp-rate duration is defined as *Ω*=*N**Δ**β*. Thus, it has the following relationship:

$$ \begin{aligned} T_{c} &= 1/f_{c},\,\, \Delta\beta=2\pi/T_{s}^{2} \\ \Omega &= 2\pi f_{c}/T_{s},\,\, \beta t^{2}=kl^{2}\frac{2\pi}{N^{2}} \\ F(k\Delta\beta)&=\frac{2}{f_{c}}\sum\limits_{l=0}^{N} f({lT}_{c}){lT}_{c}\exp\left(-j\frac{2\pi}{N^{2}}kl^{2}\right) \end{aligned} $$

The D-BFT and its inverse transform (D-IBFT) can be derived as

$$ \begin{aligned} F(k)&=\frac{2}{f_{c}}\sum\limits_{l=0}^{N} f(l)l\exp\left(-j\frac{2\pi}{N^{2}}kl^{2}\right) \\ f(l)&=\frac{1}{N^{2}}\sum\limits_{k=0}^{N}F(k)\exp\left(j\frac{2\pi}{N^{2}}kl^{2}\right) \end{aligned} $$

(3)

The BFT is shown to be equivalent to the Fourier transform of a nonlinearly sampled or warped version of the signal. Therefore, the FFT can be used to improve its computing efficiency. It can be accomplished by firstly oversampling *f*(*t*) to reduce approximation errors in obtaining warped signal samples.

By analyzing the BFT and D-BFT, we derive some characteristics and performances of the transform in multichirp-rate signal detection.

###
**Theorem 1**

The BFT impulse position is determined by the time-bandwidth product of chirp-rate signal. When the time-bandwidth product is an integer multiple of 2, the BFT impulse peak is at the discrete *k*.

###
*Proof*

Suppose chirp-rate signal is *f*(*t*)=*A* exp(*j**π**K*_{
m
}*t*^{2}). Its discrete form is \(f({lT}_{c})=A\exp \left (j\pi K_{m} l^{2}T_{c}^{2}\right) \). Inserting it into the above D-BFT expression, we obtain the peak position *κ* of *F*(*k*), which is given by

$$ \kappa=\frac{1}{2}K_{m}T_{c}^{2} N^{2}=\frac{1}{2}K_{m}T_{s}^{2} $$

(4)

According to *K*_{
m
}=±*B*_{
m
}/*T*_{
s
} where *B*_{
m
} is the bandwidth of chirp-rate signal, we have *κ*=±*B*_{
m
}*T*_{
s
}/2. Therefore, the BFT peak is at the integer *k* when the chirp *B*_{
m
}*T*_{
s
} is a multiple of 2. □

By calculating the BFT at *κ*=±*B*_{
m
}*T*_{
s
}/2 (*m*=1,2,⋯,*M*), the BFT peak of the *m*th chirp-rate signal is obtained. This operation undoubtedly simplifies the BFT of chirp-rate signal and reduces computational load.

###
**Lemma 1**

When the chirp-rate signals have different integer *B*_{
m
}*T*_{
s
}/2, they are orthogonal to each other at the discrete *k* in BFT domain.

###
*Proof*

In a symbol duration *T*_{
s
}, the BFT of *f*(*t*)=*A* exp(*j**π**K*_{
m
}*t*^{2}), (0≤*t*<*T*_{
s
}) is given by

$${} {\begin{aligned} \text{BFT}[f(t)]&=2A{\int\nolimits}_{0}^{T_{s}} t\exp\left(-j(\beta-\pi K_{m})t^{2}\right)dt \\ &={AT}_{s}^{2}\text{Sa}\left(\frac{(\beta-\pi K_{m}) T_{s}^{2}}{2}\right)\exp\left(-j(\beta-\pi K_{m}) T_{s}^{2}/2\right) \end{aligned}} $$

(5)

which is a *S**a*(·) impulse. Thus, there is

$$ |\text{BFT}[f(t)]|={AT}_{s}^{2}\text{Sa}\left(\frac{(\beta-\pi K_{m}) T_{s}^{2}}{2}\right) $$

(6)

With \(\Delta \beta \,=\,2\pi /T_{s}^{2}\), substituting \(\beta =\pi K_{m}\pm 2\pi k/T_{s}^{2}\, (k\neq 0, k=\pm 1,\pm 2,\cdots)\) into the above BFT result, it has

$$ |\text{BFT}[f(t)]|={AT}_{s}^{2}\text{Sa}\left(\frac{2k\pi}{2}\right)=0 $$

(7)

In addition, it has the peak \(|\text {BFT}[f(t)]|={AT}_{s}^{2} \) at *β*=*π**K*_{
m
}, that is *k*=*β*/*Δ**β*=*B*_{
m
}*T*_{
s
}/2 in the discrete BFT. Therefore, the chirp-rate signals have the orthogonality in discrete BFT domain if they have different integer *B*_{
m
}*T*_{
s
}/2. □

According to the lemma, the multichirp-rate symbol can be demodulated, and the BFT impulses corresponding to different chirp rates can be resolved.

###
**Theorem 2**

In the AWGN channel, SNR of the BFT detection to chirp-rate signal depends on the time-bandwidth product of the received signal where the bandwidth is the sampling bandwidth. The BER of BFT demodulation is

$$ P_{e} = Q\left(\sqrt{\frac{3}{2}\cdot\frac{E_{b}}{N_{0}}}\right) $$

(8)

where *E*_{
b
} is the received chirp-rate energy in a symbol period, and *N*_{0} is the single-sided power spectral density of AWGN.

###
*Proof*

In the continuous time domain, the peak amplitude of BFT output is \({AT}_{s}^{2}\) where A is the amplitude of the received signal. Owing to the I-Q complex modulation in Eq. (1), the peak power by BFT demodulation is given by

$$ S_{out}=A^{2} T_{s}^{4} $$

(9)

In the complex zero-mean AWGN environment, the noise can be decomposed into *n*(*t*)=*a*_{
n
}(*t*) exp(*j**ϕ*(*t*)), where *a*_{
n
}(*t*) is the amplitude with Rayleigh distribution, and *ϕ*(*t*)∈(0,2*π*] is the phase with uniform distribution, and they are statistically independent of each other [32]. Thus, the noise by BFT can be expressed as

$$ \begin{aligned} N_{\text{out}}(\beta)&=2{\int\nolimits}_{0}^{T_{s}}n(t)t\exp\left(-j\beta t^{2}\right)dt \\ &=2{\int\nolimits}_{0}^{T_{s}}t a_{n}(t)\exp\left(j\phi(t)-j\beta t^{2}\right)dt \\ &=2{\int\nolimits}_{0}^{T_{s}}t n'(t)dt \end{aligned} $$

(10)

Since *ϕ*(*t*) follows uniform distribution, (*ϕ*(*t*)−*β**t*^{2}) also follows uniform distribution. Thus, *n*^{′}(*t*)=*a*_{
n
}(*t*) exp(*j**ϕ*(*t*)−*j**β**t*^{2}) is still a Gaussian noise due to the independence of *a*_{
n
}(*t*) and (*ϕ*(*t*)−*β**t*^{2}), which obey Rayleigh distribution and uniform distribution, respectively. Then, the noise variance of the BFT can be represented as

$${} {{\begin{aligned} Var[N_{\text{out}}(\beta)]=&4\int{\int\nolimits}_{0}^{T_{s}}E\left(t_{1} n'(t_{1})t_{2} n'(t_{2})\right){dt}_{1}{dt}_{2} \\ =&4\int{\int\nolimits}_{0}^{T_{s}}t_{1}t_{2} R_{n'}(t_{1},t_{2}){dt}_{1}{dt}_{2} \\ =&4{\int\nolimits}_{-T_{s}}^{0} R_{n'}(\tau)d\tau{\int\nolimits}_{-\tau}^{T_{s}}t_{2}(\tau+t_{2}){dt}_{2} \\ &+4{\int\nolimits}_{0}^{T_{s}} R_{n'}(\tau)d\tau {\int\nolimits}_{0}^{T_{s}-\tau}t_{2}(\tau+t_{2}){dt}_{2} \\ =&4{\int\nolimits}_{-T_{s}}^{0} R_{n'}(\tau)\left(\frac{1}{2}T_{s}^{2}\tau+\frac{1}{3}T_{s}^{3}-\frac{1}{6}\tau^{3}\right)d\tau \\ &+4{\int\nolimits}_{0}^{T_{s}} R_{n'}(\tau)\left(\frac{1}{2}(T_{s}-\tau)^{2}\tau+\frac{1}{3}(T_{s}-\tau)^{3}\right)d\tau \end{aligned}}} $$

(11)

By the front band-pass filter and sampler in the receiver, *n*^{′}(*t*) is the Gaussian noise with bandwidth *B*=*f*_{
c
}. Therefore, its autocorrelation function is \(R_{n'}=\sigma _{n}^{2} S_{a}(\pi \tau /T_{c})\). We note that \(R_{n'}(\tau)\approx 0\phantom {\dot {i}\!}\) when *τ*>*T*_{
c
}. In addition, when *τ*≤*T*_{
c
}, i.e., *τ*≪*T*_{
s
}, the item \(\frac {1}{6}\tau ^{3}\) is negligible with the oversampling times *N*≫1 and \((T_{s}-\tau)^{2}\approx T_{s}^{2}\), \((T_{s}-\tau)^{3}\approx T_{s}^{3}\). Thus, we have

$${} \begin{aligned} Var[N_{\text{out}}(\beta)]&\approx 4{\int\nolimits}_{-T_{s}}^{T_{s}} R_{n'}(\tau)\left(\frac{1}{2}T_{s}^{2}\tau+\frac{1}{3}T_{s}^{3}\right) d\tau \\ &=\frac{4}{3}\sigma_{n}^{2} T_{s}^{3}{\int\nolimits}_{-T_{s}}^{T_{s}}S_{a}\left(\frac{\pi\tau}{T_{c}}\right)d\tau =\frac{4}{3}\sigma_{n}^{2}T_{s}^{3}T_{c} \end{aligned} $$

(12)

The item \(R_{n'}(\tau)\cdot T_{s}^{2}\tau \) is an odd function of *τ*, so that its integral over [−*T*_{
s
},*T*_{
s
}] is zero.

We note that the BFT in Eq. (10) is a linear transform, so that *N*_{out}(*β*) is still a Gaussian noise. Therefore, the BER of BFT demodulation for the bipolar chirp-rate signal could be derived as

$$ \begin{aligned} P_{e} &= Q\left(\sqrt{\frac{2S_{\text{out}}}{Var[N_{\text{out}}(\beta)]}}\right) \\ &=Q\left(\sqrt{\frac{3}{2}\cdot\frac{A^{2} T_{s}}{\sigma_{n}^{2} T_{c}}}\right) \end{aligned} $$

(13)

Without regarding to the processing gain *g*=*T*_{
s
}/*T*_{
c
}=*N*, the BER is expressed by \(E_{b}/N_{0}=A^{2}T_{s}/\sigma _{n}^{2}T_{c}\) as

$$ P_{e}=Q\left(\sqrt{\frac{3}{2}\cdot\frac{E_{b}}{N_{0}}}\right) $$

(14)

□

Here, we further deduce the output SNR of the D-BFT. The discretized chirp-rate signal \(f({lT}_{c})=A\exp \left (j\pi K_{m} l^{2} T_{c}^{2}\right) \) is transformed as

$$ \begin{aligned} S(k)&=\text{D-BFT}[f(t)]\\ &=\frac{2{AT}_{s}}{N}\sum\limits_{l=0}^{N} l \exp\left(j\pi K_{m} l^{2}T_{c}^{2}\right)\exp\left(-j\frac{2\pi}{N^{2}}kl^{2}\right) \end{aligned} $$

(15)

At *k*^{′}=*B*_{
m
}*T*_{
s
}/2, the amplitude of matching impulse by D-BFT is derived to be *S*(*k*^{′})=(*N*+1)*A**T*_{
s
}.

By expressing the discrete zero-mean AWGN as *n*(*l*)=*A*_{
l
} exp(*j**X*_{
l
}), where *A*_{
l
} meets Rayleigh distribution and *X*_{
l
} meets uniform distribution in (0,2*π*]. Similar to the deduction in Eq. (11), substituting *n*(*l*) into the D-BFT expression, we have

$$ \begin{aligned} N(k)&=\text{D-BFT}[n(t)]\\ &=\frac{2T_{s}}{N}\sum\limits_{l=0}^{N} l\cdot A_{l}\exp\left({jX}_{l}-j\frac{2\pi}{N^{2}}kl^{2}\right) \\ &=\frac{2T_{s}}{N}\sum\limits_{l=0}^{N} l\cdot n'(k,l) \end{aligned} $$

(16)

Shown in the expression, the output noise from the D-BFT is a linear accumulation of Gaussian processes. Obviously, it is still a zero-mean Gaussian noise, so that its variance is derived as

$$ \begin{aligned} Var[N(k)] &= E\left[\bigg(\frac{2T_{s}}{N}\sum\limits_{l=0}^{N} l\cdot n'(k,l)\bigg)^{2}\right] \\ &=\frac{4T_{s}^{2}}{N^{2}}\sigma_{n}^{2}\sum\limits_{l=0}^{N} l^{2} \\ &=\frac{2T_{s}^{2}(N+1)(2N+1)}{3N}\sigma_{n}^{2} \end{aligned} $$

(17)

Here, we have the output SNR of the D-BFT on chirp-rate signal when *N*≫1

$$ \frac{S(k')^{2}}{Var[N(k)]} = \frac{3N(N+1)}{2(2N+1)}\cdot \frac{A^{2}}{\sigma_{n}^{2}} \approx \frac{3N}{4}\cdot \frac{A^{2}}{\sigma_{n}^{2}} $$

(18)

This result is consistent with the SNR of the BFT to continuous signal, and the additional *N* in the numerator is the processing gain of the D-BFT demodulator.