84.4 The Concept of Filtering To understand the mechanism behind the design of frequency-selective filters, consider a real-coefficient ti discrete time system characterized by a magnitude function H(eJo) O≤0 0,O2<0≤兀
§4.4 The Concept of Filtering • To understand the mechanism behind the design of frequency-selective filters, consider a real-coefficient LTI discretetime system characterized by a magnitude function ( ) j H e p c c 0, 1
84.4 The Concept of Filtering We apply an input x[n= Acoso1n+ Bcos2n,0≤01<002<兀 to this system Because of linearity, the output of this system is of the form y[n]=AH(e jo1 )cos(o(n+0(01) BH(e/o2 )cos(o2n+O(02)
§4.4 The Concept of Filtering • We apply an input x[n]=Acos1n+Bcos2n, 0< 1< c< 2<p to this system • Because of linearity, the output of this system is of the form [ ] ( ) cos( ( )) 1 1 = 1 + y n AH e n j ( ) cos( ( )) 2 2 + 2 + B H e n j
84.4 The Concept of Filtering As H(e0)1.H(em)=0 the output reduces to y小=AH(em)os(on+0o) Thus, the system acts like a lowpass filter In the following example, we consider the design of a very simple digital filter
§4.4 The Concept of Filtering • As ( 1 ) 1, ( 2 ) 0 j j H e H e [ ] ( ) cos( ( )) 1 1 1 + y n AH e n j • Thus, the system acts like a lowpass filter • In the following example, we consider the design of a very simple digital filter the output reduces to
84.4 The Concept of Filtering Example- The input consists of a sum of two sinusoidal sequences of angular frequencies 0.1 rad/sample and 0.4 rad/sample We need to design a highpass filter that will pass the high-frequency component of the input but block the low-frequency component e. For simplicity, assume the filter to be an Fir filter of length 3 with an impulse response: h|0]=h2l=a,h[1=β
§4.4 The Concept of Filtering • Example - The input consists of a sum of two sinusoidal sequences of angular frequencies 0.1 rad/sample and 0.4 rad/sample • We need to design a highpass filter that will pass the high-frequency component of the input but block the low-frequency component • For simplicity, assume the filter to be an FIR filter of length 3 with an impulse response: h[0] = h[2] = a, h[1] = b
84.4 The Concept of Filtering The convolution sum description of this filter is then given by yIn=+h1xn-1+h2]] =axn +Bxn-1+ ax[n-2 yIn and xn are, respectively, the output and the input sequences Design Objective: Choose suitable values of a and B so that the output is a sinusoidal sequence with a frequency 0.4 rad/ sample
§4.4 The Concept of Filtering • The convolution sum description of this filter is then given by y[n]=h[0]x[n]+h[1]x[n-1]+h[2]x[n-2] =ax[n] +bx[n-1]+ ax[n-2] • y[n] and x[n] are, respectively, the output and the input sequences • Design Objective: Choose suitable values of a and b so that the output is a sinusoidal sequence with a frequency 0.4 rad/sample