I am trying to compute the fundamental phasor using sliding window DFT. I have employed a Blackman window in conjunction i.e $$ \sum_{k=0}^{L_{DFT}-1}x(k) w(k) e^{-j2\pi k/N} $$ where $x(k)$ is the time-domain signal sampled at a rate which is an integral multiple of the fundamental frequency i.e.$F_s$=$(f_0 \cdot N)$, where $N$ is the number of samples per cycle of the fundamental, and $w(k)$ represents the Blackman window. I am referring to a paper which is giving me an equation that should compensate for the magnitude attenuation due to this window whenever the time-domain signal frequency deviates from the nominal. The equation is
$$H_{2}(e^{j \omega_2})=0.42 H_{1}e^{j\omega_2} +0.25\left(H_{1}e^{j\left(\omega_{2}-\frac{2\pi}{L_{DFT}-1}\right)}+H_{1}e^{j\left(\omega_{2}+\frac{2\pi}{L_{DFT}-1}\right)}\right) +0.04\left(H_{1}e^{j\left(\omega_{2}-\frac{4\pi}{L_{DFT}-1}\right)}+H_{1}e^{j\left(\omega_{2}+\frac{4\pi}{L_{DFT}-1}\right)}\right).$$
The author also mentions that $$ \left\lvert H_{1} e^{(j \omega_{1})}\right\rvert=\left\lvert\frac{\sin(L_{maf}\omega_{1}/2)}{L_{maf}\sin(\omega_{1}/2)}\right\rvert$$
where $L_{maf}$ is the length of a simple moving average filter used in the pre-DFT computing process and $$ \omega_1=\frac{2\pi f_F}{F_s L_{maf}}\quad\text{also}\quad \omega_{2}=\frac{2\pi f_{F}}{F_s}$$ with $f_F$ being the computed time-domain signal frequency.
Using values such as $L_{maf}=31$, $L_{DFT}=95$, $f_0=50$, $F_s=50\cdot 24$.
The list of values obtained for $H_2$ for a list of values of $f_{F}$ are given below:
- 50 Hz=0.0639,
- 51 Hz=0.0618,
- 52 Hz=0.0594,
- 53 Hz=0.0569,
- 54 Hz=0.0543,
- 55 Hz=0.0516.
Questions:
- Why is $L_{maf}$ involved in $H_2$ computation?
- Why are the values of magnitude gain of the blackman window not accurately compensating for the magnitude attenuation caused by the same, i.e. for 50 Hz?
The gain should have been unity exactly since the DFT is centered around it, and then close to unity as the frequency deviates from the nominal.
