A signal which consists of the a sum of a large number of harmonically related sines called a multisine. Its time-domain waveform is described by the following relation:

$A_{k}$ is the amplitude of the $k^{\mathrm{th}}$ component of the multisine, $\phi_{k}$ is the phase of the $k^{\mathrm{th}}$ component. Usually, the $\phi_{k}$ are chosen randomly from $\left[0,2\pi\right[$. $f_{\mathrm{res}}$ is the base frequency of frequency resolution of the multisine. The period of the multisine is determined by $\frac{1}{f_{\mathrm{res}}}$. The maximum frequency of the multisine $k_{\mathrm{max}}f_{\mathrm{res}}$ will be denoted $f_{\mathrm{max}}$. We will make a difference between lowpass and bandpass multisines.

2.1 Bandpass multisines

In a bandpass multisine, the $A_{k}$ are zero for all frequencies smaller than $f_{\mathrm{min}}= k_{\mathrm{min}} f_{\mathrm{res}}$.

Figure 2.1 Bandpass multisine

The bandwidth of the multisine $f_{\mathrm{bandwidth}}$ is defined as $f_{\mathrm{max}}- f_{\mathrm{min}}$. The center frequency of the multisine is $f_{\mathrm{center}}=\frac{1}{2}\left( f_{\mathrm{max}}+ f_{\mathrm{min}}\right)$. To simplify simulating the multisine, we will always ensure that $f_{\mathrm{center}}$ is an integer multiple of $f_{\mathrm{res}}$: $f_{\mathrm{center}}= k_{\mathrm{center}} f_{\mathrm{res}}$. This means that $k_{\mathrm{max}}+ k_{\mathrm{min}}$ has to be an even number. Due to this limitation, the amount of tones that can fall in the bandwidth of the bandpass multisine will always be odd. This amount of tones will be indicated with $k_{\mathrm{tones}}$ and it is equal to $k_{\mathrm{max}}- k_{\mathrm{min}}+1$.

The spectral regrowth generated by a non-linear circuit excited by bandpass multisine will have the shape as shown in the figure below:

Figure 2.2 response of a nonlinear system to a bandpass multisine

The black area in the spectrum indicates the frequency band excited by the bandpass multisine. Odd order non-linear contributions are indicated in red, while even order contributions are shown in blue. The non-linear distortion appears in frequency bands around the harmonics of the center frequency. When the bandwidth is small enough and the spectral regrowth due to the non-linearity remains limited, no non-linear contributions are present in between the different bands. This property will be exploited in harmonic balance and envelope simulations. Even and odd non-linear contributions appear only around the even and odd harmonics of the center frequency respectively, so they are automatically separated in bandpass systems. The maximum bandwidth of non-linear contributions around the harmonics of the center frequency is $\aleph f_{\mathrm{bandwidth}}$. The maximum frequency at which a non-linear contribution appears is $\aleph f_{\mathrm{max}}=\aleph\left(f_{\mathrm{center}}\frac{ f_{\mathrm{bandwidth}}}{2}\right)$.

2.2 Lowpass multisines

A lowpass multisine excites frequencies starting from DC, so $f_{\mathrm{min}}= f_{\mathrm{res}}$.

Figure 2.3 Lowpass multisine

The distortion generated by a non-linear circuit as a response to a lowpass multisine is shown below:

Figure 2.4 Response of a nonlinear system to a lowpass multisine

Again, the multisine is indicated in black, even-order distortion in blue and odd-order distortion in red. With a lowpass multisine as excitation signal, the even and odd non-linear contributions will overlap. Only exciting odd frequency bins in the multisine ($A_{k}=0$ for all even $k$) allows to split the even and odd non-linear contributions: Even order non-linear distortion will be mapped on even frequency lines, while odd non-linear distortion will contribute on odd frequency lines. Note that DC should not be excited when applying the odd multisine excitation.

2.3 Multisine implementation in circuit simulation

In a circuit simulator, the multisine is best implemented with parallel current sources that inject their current in a $1\Omega$ resistor, followed by a voltage controlled voltage source with the specified output impedance [2]. If a multisine current source is required, only the parallel current sources should be added to the netlist. Both circuits are shown in the figure below:

Figure 2.5 Implementation of a multisine source

Obtaining the response of a non-linear circuit to a multisine excitation can be done with many different simulators. The popular simulators are Harmonic-Balance (HB), Envelope, Transient and Periodic Steady State (PSS) simulator. For the circuit-level simulations in my PhD thesis, Keysight’s ADS was used. In ADS, only the first three simulators are available, so these will be discussed further in this blog. The PSS simulator can be considered as a special case of the transient simulation that imposes the periodicity of the initial and the final state, and it requires similar simulation settings than a transient simulation.