# IpDFT-Tuned Estimation Algorithms for PMUs: Overview and Performance Comparison
## 2 Theoretical Background
### Tuned Frequency Down-Conversion and Low-Pass Filtering (DCF)  
As briefly outlined in the Introduction, the DCF approach is one of the most classic techniques for synchrophasor estimation. Indeed, it was mentioned in past editions of the Standard, too. The underlying idea is quite simple and it is very similar to the direct frequency down-conversion technique implemented in many radio receivers. If the samples of the \(r\)-th data record (2) are mixed with two digital quadrature sinewaves of unit amplitude and frequency \(f_{1,r}\), the following in-phase (\(l\)) and quadrature (\(Q\)) components of the input signal are obtained from the prosthaphaeresis formulas:  
\[ \begin{split} y_{r}^{I}(m)=& y_{r}(m)\cos\left(2 \pi\frac{f_{1,r}}{f_{s}}m\right)=\\ &=\frac{a_{1}(rL+m)}{2}\left[\cos(\phi_{1}(rL+m))+\cos\left(4\pi \frac{f_{1,r}}{f_{s}}m+\phi_{1}(rL+m)\right)\right]\\ &+\eta^{I}(rL+m)\\ y_{r}^{Q}(m)=&-y_{r}(m)\sin\left(2\pi\frac{f_{1,r}}{f_ {s}}m\right)=\\ &=\frac{a_{1}(rL+m)}{2}\left[\sin(\phi_{1}(rL+m))+\sin\left(4\pi \frac{f_{1,r}}{f_{s}}m+\phi_{1}(rL+m)\right)\right]\\ &+\eta^{Q}(rL+m)\end{split}  \tag{27}\]  
where \(\eta^{I}(\cdot)\) and \(\eta^{Q}(\cdot)\) are the contributions due to harmonics, inter-harmonics and wide-band noise resulting from the frequency down-conversion of term \(\eta(\cdot)\) in (2). Expression (27) shows clearly that, as a result of mixing, the power of the fundamental sinewave is evenly split into two components: one at DC, and another around \(2f_{1,r}\). Similarly, every harmonic and inter-harmonic component is split into pairs of contributions shifted by \(\pm f_{1,r}\) with respect to their respective original frequency. It is worth emphasizing that the frequency of the digital quadrature oscillators can be hardly set exactly equal to the fundamental frequency \(f_{1,r}\). Therefore, in practice, either it is set equal to the nominal value \(f_{n}\) (as suggested in the Standard) or it is extracted from the input signal. This can be done, for instance, by estimating\(f_{1,r}\) with a phase-locked loop or with a preliminary IpDFT algorithm, as done in this paper [32]. Now, let \(h(m)\) be the impulse response of a digital low-pass filter with a a null phase response and with magnitude response \(\left|H\left(e^{i\omega}\right)\right|\) close to 2 in the passband (assuming a cutoff frequency \(\omega<2\pi\frac{f_{n}}{f_{s}}\)), and to 0 in the stopband, respectively. If this filter is applied in parallel to both \(y_{r}^{I}(m)\) and \(y_{r}^{Q}(m)\), it follows that  
\[ \begin{split}\widetilde{y}_{r}^{I}(m)&=y_{r}^{I}(m )\oplus h(m)=a_{1}(rL+m)\cos(\phi_{1}(rL+m))+e_{r}^{I}(m)\\ \widetilde{y}_{r}^{Q}(m)&=y_{r}^{I}(m)\oplus h(m)=a_ {1}(rL+m)\sin(\phi_{1}(rL+m))+e_{r}^{Q}(m).\end{split}  \tag{28}\] \begin{table}
\begin{tabular}{l l} \hline \hline Step & Apply the IpDFT algorithm based on the Hann window to the \(r\)-th data record \(y_{r}(\cdot)\) and \\
**1:** & determine the estimator \(\hat{\
u}_{10,r}\) from (7). \\
**Step** & Compute the elements of the vector \(\hat{P}_{r}\) through (23). \\
**2:** & Compute the estimators \(\hat{\mu}_{1,r}\) and \(\hat{\phi}_{1,r}\) by (24), and the estimators of \(\hat{f}_{1,r}\) and \(RO\hat{CO}F_{1,r}\) with \\
**3:** & (25) and (26), respectively. \\ \end{tabular}
\end{table} 
Table 2: Steps of the tuned real-valued TWLS estimation algorithm for PMUs. In (28) (r) denotes the convolution operator, while \(e_{r}^{I}(m)\) and \(e_{r}^{Q}(m)\) are the error terms affecting the baseband I/Q components due to the filter non-idealities. These are due to (i) the frequency fluctuations of the quadrature sinewaves generated by the internal digital oscillator used for frequency down-conversion; (ii) the ripples affecting the gain of the low-pass filter in the passband; and (iii) the imperfect attenuation of both the double-frequency terms and all the disturbances lying in the stopband. However, if the filter is suitably designed, such error terms can be kept under control.