Filter Design Specifications Chaiwoot Boonyasiriwat September 29, - PowerPoint PPT Presentation

Filter Design Specifications Chaiwoot Boonyasiriwat September 29, 2020

Filter Design Specifications ▪ Lowpass filter is the most common type of digital filter. ▪ “A lowpass filter is a filter that removes the higher frequencies but passes the lower frequencies.” Example: Passband Magnitude response of a lowpass Chebyshev-I filter of order n = 4. Transition band F p / f s = 0.15,  p = 0.08 F s / f s = 0.25,  s = 0.08 Stopband 2 Schilling and Harris (2012, p. 338)

Filter Design Specifications ▪ Passband has width F p and height  p . The desired magnitude response must meet the passband specification where is the passband cutoff frequency , is the passband ripple factor (magnitude response sometimes oscillates within the passband). ▪ Stopband has width f s /2 – F s and height  s . The desired magnitude response must meet the stopband specification where is the stopband cutoff frequency , and is the stopband attenuation . 3 Schilling and Harris (2012, p. 339)

Filter Design Specifications ▪ The frequency band [ F p , F s ] between the passband and the stopband is called the transition band . ▪ For a filter to physically realizable, its passband ripple factor, stopband attenuation, and positive transition band width must be positive, i.e.,  p >0,  s > 0, | F s - F p | > 0. Example: Ideal lowpass filter has  p =  p = 0, and F s = F p . 4 Schilling and Harris (2012, p. 339)

Frequency-selective Filters ▪ Recall that a stable system with transfer function H ( z ) has the frequency response which can be written in polar form as where A ( f ) and  ( f ) are the magnitude and phase responses of the filter, respectively. ▪ The steady-state response to the sinusoidal input is Gain Phase shift 5 Schilling and Harris (2012, p. 342)

Linear Design Specifications ▪ “Most filters fall into 4 basic categories: lowpass, highpass, bandpass, and bandstop .” ▪ Magnitude responses of ideal versions of the 4 basic filter types are shown in the figure. ▪ In all case, the upper frequency limit is the Nyquist frequency f s /2 since this is the highest frequency a digital filter can process. 6 Schilling and Harris (2012, p. 343)

Linear Design Specifications ▪ The passband is the range over which A ( f ) = 1. ▪ The stopband is the range over which A ( f ) = 0. ▪ If the gain in the passband is larger than 1, the signal in the passband is amplified. ▪ Paley-Wiener Theorem: Let H ( f ) be the frequency response of a stable causal filter with A ( f ) = | H ( f )|. Then ▪ All ideal filters completely attenuate the signal over the stopband, i.e.,  s = 0. “Since log(0) = -  , it follows from the Paley-Wiener theorem that none of the ideal filters can be causal.” 7 Schilling and Harris (2012, p. 343)

Design of Practical Lowpass Filter A practical design specification for the magnitude response of a lowpass filter requires  p > 0,  s > 0, and a small transition band. The ideal cutoff frequency F p is split into two cutoff frequencies F p and F s . 8 Schilling and Harris (2012, p. 345)

Design of Practical Highpass Filter 9 Schilling and Harris (2012, p. 345)

Design of Practical Bandpass Filter 10 Schilling and Harris (2012, p. 346)

Design of Practical Stopband Filter 11 Schilling and Harris (2012, p. 346)

Linear Design Specifications: Example ▪ Consider the first-order IIR filter with transfer function whose zero is at z = -1 and pole is at z = c . ▪ For the filter to be stable, the pole must be inside the unit circle, i.e., | c | < 1. ▪ The magnitude responses of the filter at f = 0 and f = f s /2 are ▪ Here, z = exp(j2  f T ). ▪ So this is a lowpass filter. 12 Schilling and Harris (2012, p. 347)

Linear Design Specifications: Example ▪ Suppose c = 0.5. The frequency response of this IIR filter is ▪ The magnitude response of this lowpass filter is then ▪ Suppose the cutoff frequencies are F p = 0.1 f s , F s = 0.4 f s . ▪ Passband ripple satisfies 1 -  p = A ( F p ) or 13 Schilling and Harris (2012, p. 347)

Linear Design Specifications: Example Stopband attenuation satisfies 14 Schilling and Harris (2012, p. 347)

Logarithmic Design Specifications (dB) ▪ Magnitude response in the decibel or dB scale: The passband ripple in dB is A p and the stopband attenuation in dB is A s . 15 Schilling and Harris (2012, p. 348-349)

Logarithmic Design Specifications (dB) ▪ Linear to logarithmic specifications: ▪ Logarithmic to linear specifications 16 Schilling and Harris (2012, p. 349)

Logarithmic Design Specifications: Example ▪ Consider the lowpass IIR filter in the previous example ▪ The cutoff frequencies are F p = 0.1 f s , F s = 0.4 f s . ▪ The passband ripple and stopband attenuation in dB are 17 Schilling and Harris (2012, p. 349)

Logarithmic Design Specifications: Example 18 Schilling and Harris (2012, p. 349)

Linear-Phase Filters ▪ It is possible to design a filter with prescribed phase responses. ▪ Consider the analog system which delays its input by  without distortion. ▪ Its frequency response is . Thus, it is an allpass filter with A ( f ) = 1 and with a linear phase response ▪ Group delay of a system is defined as ▪ The group delay for this filter is D ( f ) =  . 19 Schilling and Harris (2012, p. 350-351)

Linear-Phase Filters ▪ A digital filter H(z) is a linear-phase filter if and only if there exists a constant  such that where F z is the set of frequencies at which A ( f ) = 0. ▪ This implies the general form of a linear phase response where  is constant and  ( f ) is piecewise constant with jump discontinuities permitted at the frequency F z at which A ( f ) = 0. ▪ A linear-phase filter can be characterized in terms of the frequency response written in the general form: 20 Schilling and Harris (2012, p. 351)

Linear-Phase Filters ▪ Here the factor A r ( f ) is real but can be positive or negative. It is referred to as amplitude response of H(z). ▪ In contrast, magnitude response A ( f ) is always positive. The relationship between A r ( f ) and A ( f ) is ▪ Consider an FIR filter ▪ “For an FIR filter, there is a symmetry condition on the coefficients that guarantees a linear phase response.” 21 Schilling and Harris (2012, p. 351)

Linear-Phase Filters: Example ▪ Consider an FIR filter of order m = 4 with the transfer function ▪ Recall that for an FIR filter, h ( k ) = b k for 0  k  m . ▪ Thus, the impulse response is ▪ In this case, h ( k ) exhibits even symmetry about the midpoint k = m /2 = 2. ▪ The frequency response in terms of  = 2  f T is 22 Schilling and Harris (2012, p. 352)

Linear-Phase Filters: Example Comparing with we see that this is a linear-phase filter with phase offset  = 0, delay  = 2 T , and the amplitude response A r ( f ) is an even function. ▪ “The even symmetry of h ( k ) about the midpoint k = m /2 is one way to obtain a linear- phase filter.” ▪ “Another approach is to use odd symmetry of h ( k ) about k = m /2. 23 Schilling and Harris (2012, p. 352)

Linear-Phase Filters “Let H ( z ) be an FIR filter of order m . Then, H ( z ) is a linear-phase filter with group delay D ( f ) = mT /2 if and only if the impulse response h ( k ) satisfies the symmetry condition 24 Schilling and Harris (2012, p. 353)

Linear-Phase Filters Impulse responses of 4 types of linear-phase filters Even order Odd order 25 Schilling and Harris (2012, p. 354)

Linear-Phase Filters “The symmetry condition that guarantees a linear phase response also imposes certain constraints on the zeros of an FIR filter.” Plus sign: filter types 1-2 (even symmetry) Minus sign: filter types 3-4 (odd symmetry) 26 Schilling and Harris (2012, p. 353-354)

Linear-Phase Filters: Example ▪ Consider a type-2 filter with even symmetry and odd order. Using the plus sign in and evaluating at z = -1 yields H (-1) = - H (-1). So, every type-2 linear-phase filter has a zero at z = -1, i.e., So, type-2 filter is a lowpass filter. ▪ Consider a type-4 filter with odd symmetry and odd order. Using the minus sign and evaluating at z = 1 yields H (1) = - H (1). So, type-4 filter has zero at z = 1, i.e., So, type-4 filter is a highpass filter. 27 Schilling and Harris (2012, p. 354)

Linear-Phase Filters: Example ▪ Consider a type-3 filter with odd symmetry and even order. Using the minus sign and evaluating at z = -1 and z = 1 yields H (-1) = - H (-1) and H (1) = - H (1), respectively. So, type-3 filter has zero at z =  1, i.e., So, type-3 filter is a bandpass filter. ▪ With the same analysis, we know that type-1 filter has no zero. So, type-1 filter is an allpass filter. 28 Schilling and Harris (2012, p. 354)