By choosing the window carefully, we can manage various trade-offs so as to maximize the filter-design quality in a given application. Window functions are always time limited. This means there is always a finite integer such that for all. The final windowed impulse response is thus always time-limited, as needed for practical implementation.
For these specifications, the function kaiserord returns a beta value of and a window length of. These values are passed to the function kaiser which computes the window function itself. The ideal bandpass-filter impulse response is computed in fir1 , and the supplied Kaiser window is applied to shorten it to length. Figure 4. Note that the upper pass-band edge has been moved to Hz instead of Hz, and the stop-band begins at Hz instead of Hz as requested. While this may look like a bug at first, it's actually a perfectly fine solution.
Therefore, the only way to achieve specs when there are multiple transition regions specified is to set the main-lobe width to the minimum transition width. For the others, it makes sense to center the transition within the requested transition region.
Under the Hood of kaiserord Without kaiserord , we would need to implement Kaiser's formula [ , 67 ] for estimating the Kaiser-window required to achieve the given filter specs:.
A similar function from [ ] for window design as opposed to filter design 5. Similarly, the filter order is estimated from stop-band attenuation and desired transition width using the empirical formula. Without the function fir1 , we would have to manually implement the window method of filter design by 1 constructing the impulse response of the ideal bandpass filter a cosine modulated sinc function , 2 computing the Kaiser window using the estimated length and from above, then finally 3 windowing the ideal impulse response with the Kaiser window to obtain the FIR filter coefficients.
The following Matlab code illustrates two different bandpass filter designs. The first different transition bands illustrates a problem we'll look at. The second equal transition bands, commented out , avoids the problem. This is a well known failure mode in FIR filter design using the Remez exchange algorithm [ , ].
It can be eliminated by narrowing the transition band, as shown in Fig. Therefore, this value is quite exact for the transition-widths of FIR bandpass filters designed by the window method using the rectangular window when the main-lobe fits entirely within the adjacent pass-band and stop-band. Use fvtool to display the frequency responses of the filters. Redesign bM using a Hann window. The 'DC-0' is optional.
Compare the magnitude responses of the Hamming and Hann designs. Redesign bW using a Tukey window. Compare the magnitude responses of the Hamming and Tukey designs. For highpass and bandstop configurations, fir1 always uses an even filter order. The order must be even because odd-order symmetric FIR filters must have zero gain at the Nyquist frequency. If you specify an odd n for a highpass or bandstop filter, then fir1 increments n by 1. Frequency constraints, specified as a scalar, a two-element vector, or a multi-element vector.
If Wn is a scalar, then fir1 designs a lowpass or highpass filter with cutoff frequency Wn. The cutoff frequency is the frequency at which the normalized gain of the filter is —6 dB. If Wn is the multi-element vector [w1 w Window, specified as a vector. If you do not specify window , then fir1 uses a Hamming window. For a list of available windows, see Windows. Normalization option, specified as either 'scale' or 'noscale'. The coefficients are sorted in descending powers of the Z-transform variable z :.
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