Use the windowbased method to design an FIR bandpass filter

Use the window-based method to design an FIR bandpass filter with the following design specifications: passband edges omega_p_1 = 0.25pi and omega_p_2 = 0.35pi, stopband edges omega_s_1 = 0.2pi and omega_s_2 = 0.375pi, and identical peak passband and stopband ripple delta_p = delta_s = 0.02. Use one of the windows listed below: Choose the window with the smallest window length to meet the design specifications. Which window should be used? What is the corresponding window length parameter M? The window selected above is to be applied to an ideal bandpass filter. What is the cutoff frequency omega_c of the ideal bandpass filter? Determine its impulse response h_d[n]. Let omega[n] denote the selected window. Then h[n] = h_d [n]omega[n] is the FIR bandpass filter resulting from the window-based design. Is the filter causal? If yes, explain why. If no, find a causal design which meets the design specifications.

Solution

In signal processing, a window function (also known as an apodization function or tapering function is a mathematical function that is zero-valued outside of some chosen interval. For instance, a function that is constant inside the interval and zero elsewhere is called a rectangular window, which describes the shape of its graphical representation. When another function or waveform/data-sequence is multiplied by a window function, the product is also zero-valued outside the interval: all that is left is the part where they overlap, the \"view through the window\".

In typical applications, the window functions used are non-negative, smooth, \"bell-shaped\" curves.Rectangle, triangle, and other functions can also be used. A more general definition of window functions does not require them to be identically zero outside an interval, as long as the product of the window multiplied by its argument is square integrable, and, more specifically, that the function goes sufficiently rapidly toward zero

RECTANGULAR window could be used for designing specifically

The Bode Plot or frequency response curve above shows the characteristics of the band pass filter. Here the signal is attenuated at low frequencies with the output increasing at a slope of +20dB/Decade (6dB/Octave) until the frequency reaches the “lower cut-off” point ƒL. At this frequency the output voltage is again 1/2 = 70.7% of the input signal value or -3dB (20 log (Vout/Vin)) of the input.

The output continues at maximum gain until it reaches the “upper cut-off” point ƒH where the output decreases at a rate of -20dB/Decade (6dB/Octave) attenuating any high frequency signals. The point of maximum output gain is generally the geometric mean of the two -3dB value between the lower and upper cut-off points and is called the “Centre Frequency” or “Resonant Peak” value ƒr. This geometric mean value is calculated as being ƒr 2 = ƒ(UPPER) x ƒ(LOWER).