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)=exp(jπK
m
t2), it has BFT[f(t)]=2πδ(β−πK
m
), where the impulse position at πK
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=lT
c
and β=kΔβ, where l and k are integers. Furthermore, the N-point symbol duration is defined as T
s
=NT
c
with N≫1, and N-point chirp-rate duration is defined as Ω=NΔβ. 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(jπK
m
t2). 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 mth 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(jπK
m
t2), (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 Sa(·) 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=πK
m
, that is k=β/Δβ=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 N0 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(jϕ(t)), where a
n
(t) is the amplitude with Rayleigh distribution, and ϕ(t)∈(0,2π] 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)−βt2) also follows uniform distribution. Thus, n′(t)=a
n
(t) exp(jϕ(t)−jβt2) is still a Gaussian noise due to the independence of a
n
(t) and (ϕ(t)−βt2), 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 Nout(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)AT
s
.
By expressing the discrete zero-mean AWGN as n(l)=A
l
exp(jX
l
), where A
l
meets Rayleigh distribution and X
l
meets uniform distribution in (0,2π]. 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.