87.1 Digital Filter Specifications Here, the maximum value of the magnitude in the passband is assumed to be unity 1/(1+e2)-Maximum passband deviation, given by the minimum value of the magnitude in the passband 1/A- Maximum stopband magnitude
§7.1 Digital Filter Specifications • Here, the maximum value of the magnitude in the passband is assumed to be unity • 1/√(1+ε2 ) - Maximum passband deviation, given by the minimum value of the magnitude in the passband • 1/A - Maximum stopband magnitude
87.1 Digital Filter Specifications For the normalized specification maximum value of the gain function or the minimum value of the loss function is o dB Maximum passband attenuation n=2090(1+e/dB For &<<1. it can be shown that max -20log0(1-28 dB
§7.1 Digital Filter Specifications • For the normalized specification, maximum value of the gain function or the minimum value of the loss function is 0 dB • Maximum passband attenuation 20log (1 2 ) max 10 p α ≅ − − δ dB • For δp<<1, it can be shown that ( ) 2 αmax = 20log10 1+ ε dB
87.1 Digital Filter Specifications In practice passband edge frequency Fand stopband edge frequency Fs are specified in Hz For digital filter design, normalized bandedge frequencies need to be computed from specifications in Hz usIng bp、2nF P=2TFT Q2πF S=2TFT T
§7.1 Digital Filter Specifications • In practice, passband edge frequency Fp and stopband edge frequency Fs are specified in Hz • For digital filter design, normalized bandedge frequencies need to be computed from specifications in Hz using F T F F F p T p T p p = π π = Ω ω = 2 2 F T F F F s T s T s s = π π = Ω ω = 2 2
872 Selection of Filter Type The transfer function H(z meeting the frequency response specifications should be a causal transfer function For iir digital filter design, the iir transfer function is a real rational function of zl M 十Dz+Dz-+…+D,z H(-) d+d,z-+d,2z2+…+d N H(z) must be a stable transfer function and must be of lowest order n for reduced computational complexity
§7.2 Selection of Filter Type • The transfer function H(z) meeting the frequency response specifications should be a causal transfer function • For IIR digital filter design, the IIR transfer function is a real rational function of z-1: N N M M d d z d z d z p p z p z p z H z − − − − − − + + + + + + + + = L L 2 2 1 0 1 2 2 1 0 1 ( ) • H(z) must be a stable transfer function and must be of lowest order N for reduced computational complexity
872 Selection of Filter Type For fir digital filter design, the fir transfer function is a polynomial in z with real coefficients: H(-)=∑n]2z n= For reduced computational complexity, degree Nof h(z must be as small as possible If a linear phase is desired, the filter coefficients must satisfy the constraint: n=thN-n
§7.2 Selection of Filter Type For reduced computational complexity, degree N of H(z) must be as small as possible • If a linear phase is desired, the filter coefficients must satisfy the constraint: h[n] = ± h[N-n] = ∑ = − N n n H z h n z 0 ( ) [ ] • For FIR digital filter design, the FIR transfer function is a polynomial in z-1 with real coefficients: