Hamming Window

The Hamming window is determined by choosing $\alpha$ in (3.17) (with $\beta\isdeftext (1-\alpha)/2$ ) to cancel the largest side lobe [101].^4.4 Doing this results in the values

$$\begin{eqnarray*} \alpha &=& \frac{25}{46} \approx 0.54\\ [5pt] \beta &=& \frac{1-\alpha}{2} \approx 0.23. \end{eqnarray*}$$

The peak side-lobe level is approximately $-42.76$ dB for the Hamming window [101].^4.5 It happens that this choice is very close to that which minimizes peak side-lobe level (down to $-43.19$ dB--the lowest possible within the generalized Hamming family) [196]:

$$\alpha = 0.53836\ldots$$ (4.19)

Since rounding the optimal $\alpha$ to two significant digits gives $0.54$ , the Hamming window can be considered the ``Chebyshev Generalized Hamming Window'' (approximately). Chebyshev-type designs normally exhibit equiripple error behavior, because the worst-case error (side-lobe level in this case) is minimized. However, within the generalized Hamming family, the asymptotic spectral roll-off is constrained to be at least $-6$ dB per octave due to the form (3.17) of all windows in the family. We'll discuss the true Chebyshev window in §3.10 below; we'll see that it is not monotonic from its midpoint to an endpoint, and that it is in fact impulsive at its endpoints. (To peek ahead at a Chebyshev window and transform, see Fig.3.31.) Generalized Hamming windows can have a step discontinuity at their endpoints, but no impulsive points.

Figure 3.10: A Hamming window and its transform.

The Hamming window and its DTFT magnitude are shown in Fig.3.10. Like the Hann window, the Hamming window is also one period of a raised cosine. However, the cosine is raised so high that its negative peaks are above zero, and the window has a discontinuity in amplitude at its endpoints (stepping discontinuously from 0.08 to 0). This makes the side-lobe roll-off rate very slow (asymptotically $-6$ dB/octave). On the other hand, the worst-case side lobe plummets to $-41$ dB,^4.6which is the purpose of the Hamming window. This is 10 dB better than the Hann case of Fig.3.9 and 28 dB better than the rectangular window. The main lobe is approximately $4\Omega_M$ wide, as is the case for all members of the generalized Hamming family ( $\beta\neq 0$ ).

Due to the step discontinuity at the window boundaries, we expect a spectral envelope which is an aliased version of a $-6$ dB per octave (i.e., a $1/\omega$ roll-off is converted to a ``cosecant roll-off'' by aliasing, as derived in §3.1 and illustrated in Fig.3.6). However, for the Hamming window, the side-lobes nearest the main lobe have been strongly shaped by the optimization. As a result, the nearly $-6$ dB per octave roll-off occurs only over an interior interval of the spectrum, well between the main lobe and half the sampling rate. This is easier to see for a larger $M$ , as shown in Fig.3.11, since then the optimized side-lobes nearest the main lobe occupy a smaller frequency interval about the main lobe.

Figure 3.11: A longer Hamming window and its transform.

Since the Hamming window side-lobe level is more than 40 dB down, it is often a good choice for ``1% accurate systems,'' such as 8-bit audio signal processing systems. This is because there is rarely any reason to require the window side lobes to lie far below the signal quantization noise floor. The Hamming window has been extensively used in telephone communications signal processing wherein 8-bit CODECs were standard for many decades (albeit $\mu$ -law encoded). For higher quality audio signal processing, higher quality windows may be required, particularly when those windows act as lowpass filters (as developed in Chapter 9).