  Two families of orthonormal tapers are proposed for multi-taper spectral
analysis: minimum bias tapers, and sinusoidal tapers $\{ \bf{v}^{(k)}\}$, where
$v_n^{(k)}=\sqrt{\frac{2}{N+1}}\sin\frac{\pi kn}{N+1}$, and $N$ is the number
of points. The resulting sinusoidal multitaper spectral estimate is
$\hat{S}(f)=\frac{1}{2K(N+1)} \sum_{j=1}^K |y(f+\frac{j}{2N+2})
-y(f-\frac{j}{2N+2})|^2$, where $y(f)$ is the Fourier transform of the
stationary time series, $S(f)$ is the spectral density, and $K$ is the number
of tapers. For fixed $j$, the sinusoidal tapers converge to the minimum bias
tapers like $1/N$. Since the sinusoidal tapers have analytic expressions, no
numerical eigenvalue decomposition is necessary. Both the minimum bias and
sinusoidal tapers have no additional parameter for the spectral bandwidth. The
bandwidth of the $j$th taper is simply $\frac{1}{N}$ centered about the
frequencies $\frac{\pm j}{2N+2}$. Thus the bandwidth of the multitaper spectral
estimate can be adjusted locally by simply adding or deleting tapers. The band
limited spectral concentration, $\int_{-w}^w |V(f)|^2 df$, of both the minimum
bias and sinusoidal tapers is very close to the optimal concentration achieved
by the Slepian tapers. In contrast, the Slepian tapers can have the local bias,
$\int_{-1/2}^{1/2} f^2 |V(f)|^2 df$, much larger than of the minimum bias
tapers and the sinusoidal tapers.
