Digital Signal Processing in Communications

# Digital Signal Processing in Communications - Design of FIR...

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Unformatted text preview: Design of FIR filter Design of FIR filters by windowing FIR filter can only be implemented in digital form. Its filter design is based on the approximation of the desired frequency response in the digital domain. Most design methods assume the filter structure to be a linear phase one. In the sequel, we emphasize on the design of lowpass filter as other types of selective filters (highpass, bandpass and bandstop) can be obtained from the lowpass filter through frequency transformation or equivalent procedure. The simplest design method is called the window method. This method starts with an ideal lowpass filter and uses a window to truncate the impulse response of the ideal lowpass filter to a finite length. The frequency response of an ideal lowpass filter is given by < = otherwise e H c j d ϖ ϖ ϖ ˆ ˆ 1 ) ( ˆ The impulse response of the ideal lowpass filter ) ˆ ( ϖ d H is given by ∫ ∫-- = = c c d e d e e H n h n j n j j d d ϖ ϖ ϖ π π ϖ ϖ ϖ π ϖ π ˆ ˆ ˆ ˆ ˆ ˆ 2 1 ˆ ) ( 2 1 ] [ n n c π ϖ ) ˆ sin( = , ∞ < < ∞- n . 1 c ϖ ˆ c ϖ ˆ- π π- ϖ ˆ ) ( ˆ ϖ j d e H Note that the impulse response of the ideal lowpass filter is noncausal and not absolutely summable (unstable filter). A simple way to define the impulse response of a lowpass filter of finite length is to truncate the ideal impulse response by a finite-length window otherwise M n M n h n h d ≤ ≤ - = ] 2 / [ ] [ M is the order of the filter and the length of the filter is M +1. The order is usually set equal to an even number in order to have integer delay M /2. Example Consider 5 . ˆ = c ϖ . The ideal impulse response is depicted in the following figure for 101 samples. 2 0 4 0 6 0 8 0 1 0 0 1 2 0- 0 . 0 5 0 . 0 5 0 . 1 0 . 1 5 0 . 2 n Sam ples The magnitude response of the frequency response of length 101 is shown as follows. ( ∑ =- M n jn e n h ˆ 10 ] [ log 20 ϖ ) The lowpass filter with length 25 is constructed by truncating the impulse response to 25 coefficients. 5 1 0 1 5 2 0 2 5- 0 . 0 4- 0 . 0 2 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 . 1 2 0 . 1 4 0 . 1 6 n sam ple The magnitude response of using 25 coefficients is shown below It is noted that both lowpass filters cannot achieve the ideal response and the first sidelobe in the stopband is just slightly below –20dB. It seems that the increase in length cannot improve the stopband gain (the gain of the first sidelobe). The filter of using longer length gives narrower transition band (the bandwidth between the passband edge and the stopband edge). The above examples can be perceived as using a rectangular window to truncate the impulse response of the ideal lowpass filter. More general windows can be applied in the truncation of the impulse response as follows ] 2 [ ] 2 [ ] [ M n w M n h n h d-- = , n = 0, 1, …, M where w [ n ] is a finite length window with nonzero values for n = - M /2,…, -1, 0, 1 ,…, M /2....
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Digital Signal Processing in Communications - Design of FIR...

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