A basic diagram of the PC process considered in this work is sketched in Figure 1. Its principle is to generate cancelling pulses at the time instants where the peaks higher than the predetermined threshold are found. The generated pulses are linearly scaled and rotated with appropriate phase shift such that after their addition the original signals have the peaks reduced to the threshold [17].

### Peak-cancelling process

To perform peak cancellation on the complex baseband signal, the target signal should be oversampled as the Nyquist-rate-sampled signals cannot correctly represent the actual amplitude of peaks of the continuous-time signals [3]. The discrete complex baseband signal *s*
_{
n
} has the general form of:

$$\begin{array}{@{}rcl@{}} s_{n}=r_{n}e^{j\theta_{n}}, \end{array} $$

((1))

where *r*
_{
n
} and *θ*
_{
n
} represent the amplitude and phase, respectively, at the *n*th time instant. Suppose that there are *N*
_{
p
} peaks that are larger than the predefined threshold *A*
_{
th
} within a given time period *T*, and let \(\phantom {\dot {i}\!}\rho _{1},\rho _{2},\cdots,\rho _{N_{p}}\) denote the corresponding successive peaks observed at the time instants \(\phantom {\dot {i}\!}n_{1},n_{2}\cdots,n_{N_{p}}\), respectively. Let *g*
_{
n
} denote the impulse response of the cancelling pulse centred at *n*=0, i.e. *g*
_{0} representing its maximum value. Then, the *i*th peak cancelling pulse at the time instant *n*
_{
i
}, where *i*∈{1,⋯,*N*
_{
p
}}, is expressed as:

$$\begin{array}{*{20}l} p^{(i)}_{n} = \left(r_{n_{i}}-A_{th}\right) g_{n-n_{i}}e^{j\theta_{n_{i}}}, \end{array} $$

((2))

where the phase is rotated by \(e^{j\theta _{n_{i}}}\) to match the phase of the corresponding complex-valued peak sample, and the amplitude is scaled by \(|r_{n_{i}}-A_{\textit {th}}|\) such that the peak value at *n*=*n*
_{
i
} is equal to *A*
_{
th
} after peak cancellation. Then, the overall signal after cancellation of the entire peaks is expressed as:

$$ \begin{aligned} \bar{s}_{n}&=s_{n}- \sum_{i=1}^{N_{p}} p^{(i)}_{n} \\ &=s_{n}- \underbrace{\left[\sum_{i=1}^{N_{p}} \left(r_{n_{i}}-A_{th}\right) e^{j\theta_{n_{i}}} g_{n-n_{i}}\right]}_{=p_{n}}, \end{aligned} $$

((3))

where *p*
_{
n
} is all the combined cancelling pulses located at the time instant *n*
_{
i
}. If we ignore the change of the amplitude and average power due to the addition of all the cancelling pulses, *A*
_{
th
} is the maximum amplitude after peak cancellation. In what follows, we refer to the corresponding PAPR determined by *A*
_{
th
} as a *target PAPR*.

### Effect of the cancelling pulse

The impulse response of the cancelling pulse *g*
_{
n
} determines the resulting OOB radiation. In general, *g*
_{
n
} should be compliant to the spectral mask of a given target standard. Suppose that *s*
_{
n
} is the oversampled version of the band-limited signal and let *J* denote the oversampling factor such that the \(\tilde {s}_{k J}\) represents the samples at the Nyquist rate for an integer *k*.

The resulting signal *s*
_{
n
} is then expressed as:

$$ \begin{aligned} s_{n}=\sum_{k} \tilde{s}_{k J}h_{n-kJ}, \end{aligned} $$

((4))

where *h*
_{
n
} is the corresponding impulse response of the pulse-shaping filter, and the summation is over the range of *k* where the impulse response has a non-negligible effect. It is worth mentioning that even though the OFDM signal is not explicitly shaped by a filter, we can still find an equivalent form [25] to represent the virtual pulse-shaping filter. Therefore, Equation 4 also applies to the conventional OFDM signals.

Now we consider the scenario where one peak *ρ*
_{
i
} is detected and subtracted by a cancelling pulse \(p_{n}^{(i)}\). It then follows that:

$$ \begin{aligned} \bar{s}_{n} &=s_{n}-p_{n}^{(i)} \\ &= \sum_{k}\tilde{s}_{kJ}h_{n-kJ}-\left[\left(r_{n_{i}}-A_{th}\right) e^{j\theta_{n_{i}}}g_{n-n_{i}}\right]. \end{aligned} $$

((5))

Suppose that the peak position is precisely given at *n*
_{
i
}=*k*
_{
i
}
*J*+*b*
_{
i
}, where *k*
_{
i
} and *b*
_{
i
} are some integers. Then, Equation 5 is rewritten as:

$$ \begin{aligned} \bar{s}_{n} &= \sum_{k}\tilde{s}_{kJ}h_{n-kJ}-\left[\left(r_{n_{i}}-A_{th}\right) e^{j\theta_{n_{i}}} g_{n-(k_{i} J + b_{i})}\right] \\ & =\sum_{k, k \neq k_{i}}\tilde{s}_{kJ}h_{n-kJ}+ \left\{ \tilde{s}_{k_{i} J} -\left(r_{n_{i}}-A_{th}\right) e^{j\theta_{n_{i}}} \right\} h_{n-k_{i} J} \\ & \qquad + \underbrace{ \left(r_{n_{i}}-A_{th}\right) e^{j\theta_{n_{i}}} \left\{ h_{n-k_{i} J} - g_{n-k_{i} J - b_{i}} \right\}}_{d^{(i)}_{n}}, \end{aligned} $$

((6))

which indicates that a proper design of *g*
_{
n
} will avoid the out-of-band emission, but we still observe that distortion component \(d^{(i)}_{n}\) will affect all the other sampling instants. This distortion component can be nullified only if *g*
_{
n
} is set equal to *h*
_{
n
} and the peak position occurs at the Nyquist point, i.e. \(h_{n-k_{i} J} =g_{n-k_{i} J - b_{i}}\).

In other words, if the cancelling pulse *g*
_{
n
} is identical to *h*
_{
n
}, then no out-of-band regrowth will occur as the signal power is confined inside the pass-band of *h*
_{
n
}. In fact, the clipping and filtering approach presented in [13] corresponds to this special case where *g*
_{
n
} is the periodic sinc function [25]. Since the periodic sinc function has non-negligible impulse response over entire OFDM symbol, it causes considerable peak regrowth. Therefore, in practice, we wish to choose *g*
_{
n
} such that its side lobe (in time domain) vanishes rapidly, and yet, its frequency response has acceptable out-of-band emission in terms of adjacent channel leakage ratio (ACLR).

### Design of the cancelling pulse

As we have seen, the impulse response of the peak-cancelling pulse *g*
_{
n
}, which is essentially a finite impulse response (FIR) filter, serves as a trade-off between the out-of-band radiation and in-band distortion. Specifically, shorter impulse response results in lower in-band distortion, but it will cause an increasing amount of out-of-band radiation that may violate the specified spectral mask. Therefore, careful design of the cancelling pulse is essential.

However, there exists no solid algorithm or closed-form deviation for finding the best cancelling pulse, and thus, exhaustive attempts are necessary to find the suitable one for a specified signal and to satisfy the design requirements. For instance, three different filters (cancelling pulses) of the same length are illustrated in Figure 2 with their respective impulse response and frequency response. Here, the windowed sinc (WS) is obtained by multiplying Kaiser window to sinc function. The raised cosine (RC) and sinc are both Nyquist filters as can be seen from the left hand of the figure. The equal ripple (ER) filter is obtained by the well-established Parks-McClellan algorithm [26] which minimizes the error in pass and stop bands by employing Chebyshev approximation. The performance of the peak cancellation based on the three cancelling pulses is demonstrated in Figure 3 using numerical simulation, where a WCDMA signal is used as its test signal. In this figure, as a practical measure for PAPR, we adopt the complementary cumulative distribution function (CCDF) of the instantaneous power normalized by its average power.

From the left hand side of Figure 3, we observe that similar PAPR performance is achieved. However, comparison of the power spectra in Figure 3 with their corresponding frequency responses in Figure 2 reveals that the frequency response of the cancelling pulses has the dominant effect on the resulting spectrum after peak cancellation.

The effects of pulse length on the in-band distortion (i.e. EVM) and out-of-band distortion (i.e. ACLR) are reported in Figure 4a,b, respectively. The measurement of EVM and ACLR follows the 3rd Generation Partnership Project (3GPP) frequency division duplexing (FDD) WCDMA downlink specification [27]. We have concluded in the last subsection that a longer pulse introduces more symbol errors to the target signal, and this is validated by Figure 4a. On the other hand, it can be easily grasped by inspection of Figure 4b that ACLR reduces with longer cancelling pulse length. However, the curves shown in the figure have some fluctuations. This can be understood by inspecting Figure 5, where the impulse response for a typical low-pass filter is shown. Truncation length of the impulse response also affects the shape of the resulting frequency response. For instance, *L*
_{1} has worse out-of-band attenuation than *L*
_{3} but may be better than *L*
_{2}, because *L*
_{2} has non-zeroes at the head and end. This heuristic observation reveals a basic clue for choosing the cancelling pulse length: the cancelling pulse should be as short as possible to minimize the distortion but should be long enough to give admissible ACLR.

It can also be observed from Figure 4a,b that the cancelling pulse generated by an equal ripple filter gives the best performance both in terms of EVM and ACLR. Better EVM performance is due to the lower side lobe in time domain, and lower ACLR is due to higher out-of-band attenuation of the cancelling pulse, as can be seen from Figure 2. It can also be seen from Figure 4a that even though the raised-cosine filter completely conforms to the pulse-shaping filter for WCDMA signal, it shows worst performance when the cancelling pulse length is short. This is mainly due to the distortion effect and the high sidelobe of the raised-cosine filter. In general, designing cancelling pulse with Parks-McClellan algorithm leads to better performance than other filter types that are frequently found in the literature.