84.4 The Concept of Filtering To understand the mechanism behind the design of frequency-selective filters, consider a real-coefficient lti discrete time system characterized by a magnitude function 0≤0 (e0)三0 10.0<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 ≅ ω ( ) j H e ω < ω ≤ π ω ≤ ω c c ,0 1
84.4 The Concept of Filtering a· We apply an input X/n=Acos@, n+ Bcos,n, 0<0<0<a<] to this system Because of linearity, the output of this system is of the form M=4()8+0) +BH(e/02 )cos(@2n+B( @2))
§4.4 The Concept of Filtering • We apply an input x[n]=Acosω1n+Bcosω2n, 0< ω1< ωc< ω2<π to this system • Because of linearity, the output of this system is of the form [ ] ( ) cos( ( )) 1 1 = 1 ω + θ ω ω y n AH e n j ( ) cos( ( )) 2 2 + 2 ω + θ ω ω B H e n j
84.4 The Concept of Filtering A H(l)=1,H(e2)=0 the output reduces to 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 jω jω H e H e [ ] ( ) cos( ( )) 1 1 ≅ 1 ω + θ ω ω 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 For simplicity, assume the filter to be an fir filter of length 3 with an impulse response:
§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] = α, h[1] = β
84.4 The Concept of Filtering The convolution sum description of this filter is then given by yIn=h(0x/n+h[1xn-1+h(2]/n-2 x]+βxn-1+axln-21 yIn and xnare, respectively, the output and the input sequences Design objective: Choose suitable values of a and p 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] =αx[n] +βx[n-1]+ αx[n-2] • y[n] and x[n] are, respectively, the output and the input sequences • Design Objective: Choose suitable values of α and β so that the output is a sinusoidal sequence with a frequency 0.4 rad/sample