Signal Toolkit - ultrwin

Function File: [w, xmu] = ultrwin (m, mu, beta)

Function File: [w, xmu] = ultrwin (m, mu, att, "att")

Function File: [w, xmu] = ultrwin (m, mu, latt, "latt")

Function File: w = ultrwin (m, mu, xmu, "xmu")

Return the coefficients of an Ultraspherical window of length m. The parameter mu controls the window’s Fourier transform’s side-lobe to side-lobe ratio, and the third given parameter controls the transform’s main-lobe width/side-lobe-ratio; normalize w such that the central coefficient(s) value is unitary.

By default, the third parameter is beta, which sets the main lobe width to beta times that of a rectangular window. Alternatively, giving att or latt sets the ripple ratio at the first or last side-lobe respectively, or giving xmu sets the (un-normalized) window’s Fourier transform according to its canonical definition:

              (MU)
      W(k) = C   [ XMU cos(pi k/M) ],  k = 0, 1, ..., M-1,
              M-1
 

where C is the Ultraspherical (a.k.a. Gegenbauer) polynomial, which can be defined using the recurrence relationship:

       (l)    1                  (l)                    (l)
      C (x) = - [ 2x(m + l - 1) C   (x) - (m + 2l - 2) C   (x) ]
       m      m                  m-1                    m-2

                                 (l)        (l)
      for m an integer > 1, and C (x) = 1, C (x) = 2lx.
                                 0          1
 

For given beta, att, or latt, the corresponding (determined) value of xmu is also returned.

The Dolph-Chebyshev and Saramaki windows are special cases of the Ultraspherical window, with mu set to 0 and 1 respectively. Note that when not giving xmu, stability issues may occur with mu <= -1.5. For further information about the window, see

• Kabal, P., 2009: Time Windows for Linear Prediction of Speech. Technical Report, Dept. Elec. & Comp. Eng., McGill University.
• Bergen, S., Antoniou, A., 2004: Design of Ultraspherical Window Functions with Prescribed Spectral Characteristics. Proc. JASP, 13/13, pp. 2053-2065.
• Streit, R., 1984: A two-parameter family of weights for nonrecursive digital filters and antennas. Trans. ASSP, 32, pp. 108-118.

See also: chebwin, kaiser.