This section introduces the main methods used in the compressive OFDM model. The generation method of compressive OFDM signal is proposed based on the band-limited characteristic of traditional OFDM signal. Accordingly, the recovery method is given with analysing the error considering influential parameters and noise.

### 3.1 Generation method of compressive OFDM signal at transmitter

An *N*-subcarrier OFDM signal with rectangular window function shaping filter can be expressed as

$$\begin{aligned} s(t) = \frac{1}{{\sqrt{T} }}\sum \limits _{n = 0}^{N - 1} {{x_n}{e^{\frac{{j2\pi nt}}{T}}}} , \end{aligned}$$

(14)

where *T* is the symbol period of OFDM signal and the subcarrier frequency space \(\Delta f = {1 / T}\) satisfying the Nyquist orthogonality criterion. As the subcarrier waveform is the sum of a set of *sinc* functions, the bandwidth can be approximately calculated as \({B_{\mathrm{OFDM}}} = N/T\). It is found that the OFDM signal can be regarded bandlimited with finite energy. According to the explanation of GP algorithm, the entire OFDM signal could be theoretically extrapolated from a segment of it. On this basis, the compressive OFDM system can be established as follows.

Figure 1 depicts the generation diagram of compressive OFDM signal based on the structure of traditional OFDM system. To obtain the compressive OFDM signal with compressed ratio \(\alpha\), the traditional time-domain OFDM signal is processed by a truncation filter. In particular, the location of the reserved original OFDM signal is defined as the support set. The generation process is equivalent to the point product of the source and the truncation filter, defined as

$$\begin{aligned} {s^\alpha }(t) = s(t) \cdot {p_\alpha }(t), \end{aligned}$$

(15)

where \(s^{\alpha }(t)\) denotes the compressive OFDM signal, and \(p_{\alpha }(t)\) is the truncation filter in the time domain satisfying

$$\begin{aligned} {p_\alpha }(t) = \hbox {rect}(\frac{t}{T}) = \left\{ {\begin{array}{ll} 1,&{}\quad t \in T(\Psi ),\\ 0,&{}\quad t \notin T(\Psi ), \end{array}} \right. \end{aligned}$$

(16)

where the rectangular window *rect*(‘) retains the data in the supporting set \(\Psi\) . The expression of the compression ratio is calculated by \(\alpha = \sum {T(\Psi )} /T\).

Obviously, information beyond the supporting set in original OFDM is discarded via truncation filter. Whereas the truncated segment is relevant to the remaining segment (the compressed OFDM symbol) via DFT so that it still can be recovered. The relationship among the remaining segment, truncated segment and the original signal can be mathematically explained in Sect. 2.2. It is found that the entire bandlimited continuous OFDM signal and the truncated compressive OFDM signal can be expressed at the same prolate sphere basis. The former one can be determined by the latter one according to the analytical continuation principle [32]. In addition, the frequency spacing of compressive OFDM signals maintains while the symbol duration turns shorter by the compressed ratio \(\alpha\), namely relationship of the symbol period of compressive OFDM and OFDM is \({T_{\mathrm{cps}}} = \alpha T\).

### 3.2 Recovery method for compressive OFDM signal at receiver

In Fig. 2, the receiver structure can be illustrated on basis of OFDM system, with an extra extrapolator module correspondingly to recover the received signal.

The signal after S/P should be up-sampled with the sample period \(T_{\mathrm{cps}}/(\alpha k N)\) , where *k* represents the up-sampling rate. The up-sampling is required due to the constraint of recovery process based on GP algorithm. Take an *N*-point information source as an example, only *N*/2-point signal is received and used to be extrapolated to obtain the following *N*/2 points if the minimum Nyquist sampling rate is considered. As the description in Table 1, the *N*/2-point received compressive OFDM time domain signal is zero-tapped to N points at the end, and then undergoes *N*-DFT, low-pass filter, and *N*-IDFT. After that, the *N*/2 points of the intermediate time domain result in the supporting set are replaced with those of received segment.

According to the description of the extrapolation filter, the low-pass filter retains all *N* points and the points beyond the supporting set are still zero after IDFT. In that case, the above-mentioned iteration is invalid and no extra point is extrapolated. Therefore, up-sampling is necessary for the recovery for compressive OFDM signal. The influence of module parameters will be discussed in 3.3.

### 3.3 Parameter setting for compressive OFDM system

#### 3.3.1 Compressed ratio and truncation filter at transmitter

Note that the compressed ratio \({\alpha }\) and truncation filter \(p_{\alpha }(t)\) should be designed in advance to determine a compressive OFDM signal. The mapping mode is BPSK and the bandwidth of extrapolation filter is the same as that of BPSK-OFDM signal. Three modes of truncation filter were compared as well, namely front segment, mid segment and back segment. Monte Carlo simulation on recovering compressive OFDM signals was carried out with different compressed ratios or truncation filters based on GP algorithm. The step size of compressed ratios was 0.01, and the maximum number of iterations was 100. The BER results were shown in Fig. 3.

It can be seen from Fig. 3 that the truncation filters remaining the front or back segment possesses comparable BER performance but better than that of mid truncation filter. For the front/back mode, the BER was relatively stable when the compressed ratio was not less than 0.50 when only considering the self-interference caused by the truncation process at the transmitter. Fluctuations of BER at \(10^{-5}\) level were negligible when the compressed ratio ranged within 0.54–0.73. In that case, the minimum compressed ratio is appropriate to be implicated as 0.5 for compressive OFDM transmission scheme. It also implies a stronger resistance to self-interference in the generation process of compressive OFDM signals compared to SEFDM.

#### 3.3.2 Bandwidth of extrapolation filter

As mentioned in Table 1, the bandwidth of the extrapolation filter is equal to that of the original function in terms of GP algorithm. With the increasing iterations, the error decreases and tends zero when the iterations tend infinite [28]. The *rect* function shaping filter is generally considered when analysing baseband OFDM signals, but the shaping filter could also be designed as Root Raised Cosine Filter (RRC) or other filters to improve the out-of-band radiation. Even so, the actual signal is not bandlimited strictly in the frequency domain. In terms of the recovery for the discrete signal, the length of the extrapolation filter reflects the filter bandwidth which is related to the points number of information source and upsampling rate. If the bandwidth of the extrapolation filter is selected as the cut-off bandwidth of the original OFDM symbol, possible aliasing error might be generated.

To evaluate the influence of the extrapolation filter bandwidth, the accuracy of extrapolating compressive OFDM with Rect shaping filter with different bandwidth is simulated, respectively. For BPSK-OFDM signal with 256 subcarriers, the compressed ratio is 0.5 and the received signal is up-sampled by 8 times before extrapolation. The up-sampled signal is recovered as revealed in Fig. 2. Normalized mean square error (NMSE) is adopted to measure the accuracy performance, expressed as

$$\begin{aligned} {\mathrm{NMSE}} = \frac{{{{\left\| {y - r} \right\| }^2}}}{{{{\left\| r \right\| }^2}}}. \end{aligned}$$

(17)

Defining the original signal points as \(\omega\) which equals to the product of the number of subcarriers and up-sample times and the low-pass filter points as \(\delta\), the NMSE values of the extrapolation by iteration filters with different bandwidth are illustrated in Fig. 4.

As shown in Fig. 4, iteration filter with the same bandwidth as that of the original signal obtains the best accuracy. The convergence of extrapolation turns slower if the filter is set with larger bandwidth, while the NMSE increases obviously with iterating when the filter is set with narrow bandwidth.

To explain with mathematical mechanism, the entire frequency domain signal could be divided into two parts as

$$\begin{aligned} S(f) = L(f) + H(f), \end{aligned}$$

(18)

where *L*(*f*) denotes the low-frequency signal at \(\left| f \right| \le \omega\), and *H*(*f*) denotes the high-frequency signal at \(\left| f \right| > \omega\) . The corresponding time domain signal can be converted into

$$\begin{aligned} s(t) = l(t) + h(t). \end{aligned}$$

(19)

According to the linear additivity of the extrapolation, the result of *m*th iteration can be expressed as

$$\begin{aligned} {r_m}(t) = {l_m}(t) + {h_m}(t). \end{aligned}$$

(20)

The convergence of the \(l_{m}(t)\) is determined by the bandlimited characteristics of *l*(*t*) and the upper bound of the energy of \(h_{m}(t)\) that could be inferred as

$$\begin{aligned} {E_{{h_m}}} < {m^2}{E_h}, \end{aligned}$$

(21)

where \(E_{h_m}\) and \(E_h\) denote the energy of \(h_{m}(t)\) and *h*(*t*), respectively. Note that the upper bound would increase with the number of iterations *m* by the square times. However, the actual value is much smaller than the upper bound. When the out-of-band energy accounts for a small proportion of the total signal energy, the extrapolation error caused can be reduced [30], which might be attributed the considerable accuracy of \(\delta / \omega = 1\) .

#### 3.3.3 Regularized extrapolator

In addition to the errors caused by the extrapolation coefficients, the received signal also contains channel noise interference, which can be expressed as

$$\begin{aligned} r(t) = {s^\alpha }(t) + z(t), \end{aligned}$$

(22)

where \(s^\alpha\) represents the compressive OFDM signal with the compressed ratio 0.5 and the original symbol period *T*. *z*(*t*) is Gaussian white noise with mean value 0 and variance \(\delta ^2\) . The signal obtained in the *m*th iteration can be described as

$$\begin{aligned} {r_m}(t) = s_m^\alpha (t) + {z_m}(t). \end{aligned}$$

(23)

At that time, adopting the iterative filter with a bandwidth of \(\delta / \omega =1\) and performing K–L expansion on *z*(*t*) in the frequency domain, the extrapolated result of Gaussian distributed *z*(*t*) in the first iteration can be obtained as

$$\begin{aligned} {z_1}(t) = B\frac{{\sin \delta t}}{{\pi t}} \otimes \sum \limits _{k = 0}^\infty {{d_k}\sqrt{{\lambda _k}} {\phi _k}(bt)} , \end{aligned}$$

(24)

where \(B = \sqrt{\frac{{\pi T}}{\delta }}\) and \(b = \frac{T}{{2\delta }}\); \(\lambda _k\) and \(\Phi _k\) are the eigenvalues and corresponding eigenfunctions of the K–L expansion. \(d_k\) denotes the expansion coefficient, satisfying

$$\begin{aligned} E\left\{ {{d_k}d_r^*} \right\} = \left\{ {\begin{array}{ll} S/{\lambda _k},&{}\quad k = r,\\ 0,&{}\quad k \ne r. \end{array}} \right. \end{aligned}$$

(25)

To simplify the convolution process of (25), the upper bound of the energy of \(z_{1}(t)\) is calculated in the frequency domain as

$$\begin{aligned} E\left\{ {{{\left| {{Z_1}(f)} \right| }^2}} \right\} = 2S\pi b\sum \limits _{k = 0}^\infty {\phi _k^2(bf)} = 2ST, \end{aligned}$$

(26)

where \(S = {\delta ^2}\Delta T\) and \(\Delta T\) represents the sampling interval.

Due to the full-band Gaussian white noise, *Zm*(*f*) still satisfies the Gaussian distribution. The mean square value of the energy of *Zm*(*f*) can be expressed as

$$\begin{aligned} E\left\{ {{{\left| {{Z_m}(f)} \right| }^2}} \right\}= & {} 2S\pi b\sum \limits _{k = 0}^\infty {{{\left[ {\frac{{1 - {{(1 - {\lambda _k})}^n}}}{{{\lambda _k}}}} \right] }^2}\phi _k^2(bf)} \\= & {} 2S\pi b\sum \limits _{k = 0}^\infty {{{(1 + {\varsigma _k}+ \cdots +\varsigma _k^{n - 1})}^2}\phi _k^2(bf)} , \end{aligned}$$

(27)

where \({\varsigma _k} = 1 - {\lambda _k}\) and \({\lambda _k} \in (0,1)\).

Inspired by [28], the upper bound of \(E\left\{ {{{\left| {{Z_m}(f)} \right| }^2}} \right\}\) can be inferred to \(2ST{n^2}\) , similar to the result of (21). Nevertheless, the difference is that the noise is not correlated with the original signal and the actual impact of the extrapolating noise will accumulate with iterating. In that case, the extrapolation process turns into an ill-posed problem.

According to the GP algorithm, the extrapolation process can be expressed as

$$\begin{aligned} {\hat{S}} = A \cdot {S^\alpha }, \end{aligned}$$

(28)

where the operator *A* contains the FT, filtering, IFT and exchange process. So is the recovery method for compressive OFDM expressed like that. Combining the idea of Landweber algorithm [33], the regularized recovery algorithm for compressive OFDM signal and the related receiver structure are shown in Fig. 5.

In *m*th iteration, the extrapolation calculation before replacement can be described as

$$\begin{aligned} {y}_m^{\tau ,\mu } = {{r}_{m - 1}} + \mu \left( {{A^*}2{r} - {A^*}A{{r}_{m - 1}} - \tau {r_{m - 1}}} \right) , \end{aligned}$$

(29)

where the regularized operator \(A = {F^{-1}}\mathbf{H }_{\omega }F\), and *F* as well as \(F^{-1}\) are DFT matrix and IDFT matrix, respectively. \(\tau\) and \(\mu\) are two parameters related to regularization, which are defined as regularized parameter and relaxation coefficient, respectively. When the extrapolator meets the judgement of iteration threshold, the iterative result \(r_m\) would be output as the recovered *y*.