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

Chaiwoot Boonyasiriwat

September 29, 2020

Filter Design Specifications

Filter Design Specifications

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▪ 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: Magnitude response of a lowpass Chebyshev-I filter of order n = 4. Fp/fs = 0.15, p = 0.08 Fs/fs = 0.25, s = 0.08

Schilling and Harris (2012, p. 338)

Passband Stopband Transition band

Filter Design Specifications

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▪ Passband has width Fp 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 fs/2 – Fs and height s. The desired magnitude response must meet the stopband specification where is the stopband cutoff frequency, and is the stopband attenuation.

Schilling and Harris (2012, p. 339)

Filter Design Specifications

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▪ The frequency band [Fp, Fs] 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, |Fs - Fp| > 0. Example: Ideal lowpass filter has p = p = 0, and Fs = Fp.

Schilling and Harris (2012, p. 339)

Frequency-selective Filters

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▪ 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

Schilling and Harris (2012, p. 342)

Gain Phase shift

Linear Design Specifications

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▪ “Most filters fall into 4 basic categories: lowpass, highpass, bandpass, and bandstop.” ▪ Magnitude responses

• f ideal versions of

the 4 basic filter types are shown in the figure. ▪ In all case, the upper frequency limit is the Nyquist frequency fs/2 since this is the highest frequency a digital filter can process.

Schilling and Harris (2012, p. 343)

Linear Design Specifications

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▪ 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.”

Schilling and Harris (2012, p. 343)

Design of Practical Lowpass Filter

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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 Fp is split into two cutoff frequencies Fp and Fs.

Schilling and Harris (2012, p. 345)

Design of Practical Highpass Filter

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Schilling and Harris (2012, p. 345)

Design of Practical Bandpass Filter

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Schilling and Harris (2012, p. 346)

Design of Practical Stopband Filter

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Schilling and Harris (2012, p. 346)

Linear Design Specifications: Example

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▪ 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 = fs/2 are ▪ Here, z = exp(j2 f T). ▪ So this is a lowpass filter.

Schilling and Harris (2012, p. 347)

Linear Design Specifications: Example

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▪ 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 Fp = 0.1fs, Fs = 0.4fs. ▪ Passband ripple satisfies 1 - p = A(Fp) or

Schilling and Harris (2012, p. 347)

Linear Design Specifications: Example

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Stopband attenuation satisfies

Schilling and Harris (2012, p. 347)

Logarithmic Design Specifications (dB)

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▪ Magnitude response in the decibel or dB scale:

Schilling and Harris (2012, p. 348-349)

The passband ripple in dB is Ap and the stopband attenuation in dB is As.

Logarithmic Design Specifications (dB)

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▪ Linear to logarithmic specifications: ▪ Logarithmic to linear specifications

Schilling and Harris (2012, p. 349)

Logarithmic Design Specifications: Example

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▪ Consider the lowpass IIR filter in the previous example ▪ The cutoff frequencies are Fp = 0.1fs, Fs = 0.4fs. ▪ The passband ripple and stopband attenuation in dB are

Schilling and Harris (2012, p. 349)

Logarithmic Design Specifications: Example

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Schilling and Harris (2012, p. 349)

Linear-Phase Filters

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▪ 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 ) = .

Schilling and Harris (2012, p. 350-351)

Linear-Phase Filters

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▪ A digital filter H(z) is a linear-phase filter if and only if there exists a constant  such that where Fz 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 Fz at which A( f ) = 0. ▪ A linear-phase filter can be characterized in terms of the frequency response written in the general form:

Schilling and Harris (2012, p. 351)

Linear-Phase Filters

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▪ Here the factor Ar( 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 Ar( 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.”

Schilling and Harris (2012, p. 351)

Linear-Phase Filters: Example

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▪ Consider an FIR filter of order m = 4 with the transfer function ▪ Recall that for an FIR filter, h(k) = bk 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

Schilling and Harris (2012, p. 352)

Linear-Phase Filters: Example

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Comparing with we see that this is a linear-phase filter with phase offset  = 0, delay  = 2T, and the amplitude response Ar( 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.

Schilling and Harris (2012, p. 352)

Linear-Phase Filters

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“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

• nly if the impulse response h(k) satisfies the symmetry

condition

Schilling and Harris (2012, p. 353)

Linear-Phase Filters

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Impulse responses of 4 types of linear-phase filters

Schilling and Harris (2012, p. 354)

Even order Odd order

Linear-Phase Filters

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“The symmetry condition that guarantees a linear phase response also imposes certain constraints on the zeros of an FIR filter.”

Schilling and Harris (2012, p. 353-354)

Plus sign: filter types 1-2 (even symmetry) Minus sign: filter types 3-4 (odd symmetry)

Linear-Phase Filters: Example

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▪ Consider a type-2 filter with even symmetry and odd

• rder. 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

• rder. 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.

Schilling and Harris (2012, p. 354)

Linear-Phase Filters: Example

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▪ Consider a type-3 filter with odd symmetry and even

• rder. 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.

Schilling and Harris (2012, p. 354)

Linear-Phase Filters

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▪ Let z = r exp(j) be a complex zero with r > 0. ▪ Then from , must also be a zero of H(z). ▪ If the coefficients bk are real, the zeros must occur in complex-conjugate pairs. Hence, for r  0, the zeros will appear in groups of 4 and satisfy the reciprocal symmetry: ▪ When the zeros are real, the set Q reduces to a pair of reciprocal real zeros.

Schilling and Harris (2012, p. 354-355)

Linear-Phase Filters: Example

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▪ A pole-zero plot for a type-1 FIR filter. ▪ All poles of FIR filter are at the origin. ▪ Zeros are distributed according to the reciprocal symmetry set.

Schilling and Harris (2012, p. 355)

Zero-Phase Filters

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▪ If the frequency response H( f ) is real and non- negative, its imaginary part is zero. So H(z) is a zero- phase filter: ▪ A zero-phase filter is a linear-phase filter with a group delay of zero, and cannot be realized with a causal system. ▪ Recall the time-reversal property of DFT: where xp(k) is the periodic extension of x(k). ▪ The time-reversed periodic extension of x(k) is x(N-k).

Schilling and Harris (2012, p. 356)

Zero-Phase Filters

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▪ Suppose F(z) is a linear-phase FIR filter of order m. Then, F(z) has frequency response F(i) and a group delay of D( f ) = mT/2. ▪ Let x(k) and q1(k) be the input and output of the filter. ▪ Then in the frequency domain, ▪ Suppose the output q1(k) is time reversed to produce a new signal q2(k) = q1(N - k). ▪ Using the time reversal property of DFT yields ▪ q2(k) is then used as an input to a second copy of F(z). ▪ The resulting output is

Schilling and Harris (2012, p. 356)

Zero-Phase Filters

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▪ q3(k) is then time reversed to produce the output y(k). ▪ So we have ▪ The noncausal system H(i) has the real non-negative frequency response. So it is a zero-phase filter. ▪ Since the FIR filter F(z) processes the signal twice, the noncausal filter H(z) is of order 2m.

Schilling and Harris (2012, p. 356-357)

Zero-Phase Filters: Example

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Consider an input signal with F0 = 20 Hz, F1 = 60 Hz, and fs = 200 Hz: Let F(z) be a lowpass FIR filter of order m = 30 with cutoff frequency

Schilling and Harris (2012, p. 357)

The noncausal filter H(z) should block the 60-Hz signal while passes the 20- Hz signal without distortion or phase shift. The ending transient is the startup transient of second stage of F(z).

Minimum-Phase and Allpass Filters

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▪ “Every digital filter with a rational transfer function can be expressed as a product of a minimum-phase filter and an allpass filter.” ▪ “The magnitude response, by itself, does not provide enough information to completely specify a filter.” ▪ “For example, among IIR filters having m zeros, there are up to 2m distinct filters, each having an identical magnitude response A( f ). The difference between these filters lie in their phase responses ( f ).” ▪ “A digital filter H(z) is a minimum-phase filter if and

• nly if all of its zeros lie inside or on the unit circle.

Otherwise, it is a nonminimum-phase filter.”

Schilling and Harris (2012, p. 359)

Minimum-Phase and Allpass Filters

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▪ “Every IIR filter H(z) can be converted to a minimum- phase filter with the same magnitude response by replacing the zeros outside the unit circle with their reciprocals.” ▪ “The term minimum phase arises from the fact that the net phase change of a minimum-phase filter, over the frequency range [0, fs/2], is .” ▪ “Nonminimum-phase filters have at least one zero

• utside the unit circle.”

▪ “If H(z) has p zeros outside the unit circle, the net phase change is ” (Proakis and Manolakis, 1988).

Schilling and Harris (2012, p. 359-360)

Minimum-Phase Filters: Example

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▪ Consider the second-order IIR filter ▪ This is a stable IIR filter with poles at z = 0.5  j0.5. ▪ It is also a minimum-phase filter because both zeros at z =  0.5 are inside the unit circle. ▪ The other 3 filters with the same magnitude response can be obtained by using the reciprocal of the first zero, the second zero, and both multiplying by the negative of the original zero.

Schilling and Harris (2012, p. 360)

Minimum-Phase Filters: Example

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▪ The transfer function of the other filters are ▪ H11(z) is a maximum-phase filter since all of its zeros are outside the unit circle. H01(z) and H10(z) are mixed- phase filters since one of their zeros are outside the unit circle.

Schilling and Harris (2012, p. 360)

Minimum-Phase Filters: Example

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Pole-zero plot of the 4 filters

Schilling and Harris (2012, p. 360)

Minimum-Phase Filters: Example

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Magnitude and phase responses of the 4 filters ▪ Net phase change of H00 is zero ▪ Net phase change of H01 and H10 is - ▪ Net phase change of H11 is -2

Schilling and Harris (2012, p. 361)

Allpass Filters

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▪ Allpass filters are IIR filters that has a flat magnitude

• response. So, all spectral components are passed equally

▪ A digital filter H(z) is an allpass filter if and only if it has the magnitude response ▪ The transfer function coefficients of allpass filters has the reflective structure

Schilling and Harris (2012, p. 362)

Allpass Filters

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▪ To see how the reflective structure give a flat magnitude response, let A( f ) denote the magnitude response of the FIR filter H(z) = a(z-1) corresponding to the denominator of the allpass filter. ▪ The magnitude response of the allpass filter is

Schilling and Harris (2012, p. 362)

Allpass Filters

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▪ “The process of converting a filter H(z) to minimum- phase form can be thought of as multiplication of H(z) by a transfer function F(z).” ▪ “Suppose H(z) has a single zero at z = c where c lies

• utside the unit circle.”

▪ “Then replacing this zero with one at z = c-1 and multiplying by –c is equivalent to multiplying H(z) by” ▪ “If z = c is the only zero of H(z) outside the unit circle, then the minimum-phase version of H(z) can be expressed as”

Schilling and Harris (2012, p. 362)

Allpass Filters

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▪ F(z) can be rewritten as whose coefficients has the reflective structure. So, F(z) is an allpass filter with a = [1, -c]T. ▪ If H(z) of multiple zeros outside the unit circle, F(z) used to convert H(z) to a minimum-phase filter is always an allpass filter. ▪ If F(z) is an allpass filter, then Hall (z) = F-1(z) is also an allpass filter since “the magnitude response of the inverse of a filter is the inverse of the magnitude response of the filter.”

Schilling and Harris (2012, p. 363)

Allpass Filters

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▪ Multiplying both side of by Hall(z) yields ▪ So, every digital filter with a rational transfer function can be expressed as a product of a minimum-phase filter and an allpass filter. ▪ Minimum-phase Allpass Decomposition Let H(z) be a rational IIR transfer function, and Hmin(z) be the minimum-phase form of H(z), Then there exists a stable allpass filter Hall(z) such that

Schilling and Harris (2012, p. 363)

Block diagram of the decomposition

Allpass Filters

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▪ An allpass filter can also be characterized by its poles and zeros. “For each pole of Hall(z) at z = c, there is a matching zero at its reciprocal z = c-1. Thus, allpass filters always have the same number of poles and zeros, with the poles and zeros forming reciprocal pairs.” Algorithm for minimum-phase allpass decomposition:

• 1. Set Hmin(z) = H(z), and Hall(z) = 1. Factor the numerator

polynomial of H(z) as

• 2. For i = 1 to m

if |zi| > 1

Schilling and Harris (2012, p. 364)

Minimum-phase Allpass Decomposition: Example

47

Consider the stable IIR filter with real poles at p1,2 = 0.8 and complex-conjugate zeros at z1,2 = -0.5  j1.5. Since both zeros are outside the unit circle, this is a maximum-phase filter.

Schilling and Harris (2012, p. 364)

Minimum-phase Allpass Decomposition: Example

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The minimum-phase form can be obtained by replacing the zeros by their reciprocals and multiplying by the negative of the zeros. The new zeros are The product z1z2 = |z1|2. Thus, the minimum-phase form is The allpass filter is

Schilling and Harris (2012, p. 365)

Minimum-phase Allpass Decomposition: Example

49

Schilling and Harris (2012, p. 365)

Quadrature Filters: Differentiator

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▪ “A pair of periodic signals is in phase quadrature if one signal leads or lags the other by a quarter of a cycle.” ▪ Example of quadrature pair: sin and cosine ▪ Some filters have steady-state outputs that are in phase quadrature with the input, e.g., differentiator. ▪ “A continuous-time differentiator has transfer function Ha(s) = s and frequency response Ha( f ) = j2f.” ▪ “To approximate the differentiator with a causal mth-

• rder linear-phase filter, consider a discrete-time

frequency response with a delay mT/2.”

Schilling and Harris (2012, p. 367)

Quadrature Filters: Differentiator

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▪ “In feedback control, a PID controller is often implemented by approximating the derivative dx(t)/dt at t = kT with a first-order backward Euler difference:” ▪ The FIR transfer function of the backward Euler differentiator is whose pole and zero are at z = 0 and z = 1, respectively.

Schilling and Harris (2012, p. 367)

Quadrature Filters: Differentiator

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▪ The frequency response in this case is ▪ “Since the backward Euler approximation is a linear- phase FIR filter of order m = 1, it includes a delay of half a sample (T/2).” ▪ The backward Euler impulse response is which exhibits odd symmetry about the midpoint m/2. ▪ “Since H(z) is of odd order, the backward Euler differentiator is a type-4 linear-phase FIR filter with a phase offset of  = /2 needed for a differentiator.”

Schilling and Harris (2012, p. 368)

Backward Euler vs Ideal Differentiator

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Schilling and Harris (2012, p. 368)

Quadrature Filters: Differentiator

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▪ Higher-order type-4 linear-phase FIR filters can approximate the ideal differentiator with a more accurate magnitude response. ▪ FIR differentiator filter of order m = 5 is ▪ This filter has a delay of 2.5 samples and a phase offset

• f  = /2.

▪ The coefficients also have the odd symmetry.

Schilling and Harris (2012, p. 368)

5th-order FIR Filter vs Ideal Differentiator

55

Schilling and Harris (2012, p. 368)

Quadrature Filters: Hilbert Transformer

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▪ Hilbert transformer is another filter that produces a steady-state output in phase quadrature with a sinusoidal input. ▪ The frequency response of Hilbert transformer is where sgn is the signum or sign function defined as ▪ Its magnitude response is |Hd( f )| = 1 and there is a constant phase shift of

Schilling and Harris (2012, p. 369)

Quadrature Filters: Hilbert Transformer

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▪ Hilbert transformer cannot be realized by a causal system since its impulse response is ▪ “To approximate the Hilbert transformer by a causal linear-phase FIR filter of order m, we include a delay of mT/2 in the desired frequency response” ▪ Type-3 or type-4 linear-phase filters can be used to approximate the Hilbert transformer.

Schilling and Harris (2012, p. 369)

Type-3 FIR filter vs Ideal Hilbert Transformer

58

Schilling and Harris (2012, p. 370)

Type-3 FIR filter of order m = 40 used here is a bandpass filter instead of an allpass filter like the ideal Hilbert transformer since it has zeros at H(1) = 0.

Quadrature Filters: Hilbert Transformer

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▪ Suppose the input is a cosine of frequency f : ▪ The total phase shift of the Hilbert transformer Hh( f ) is ▪ Suppose x2(k) is produced by passing x(k) through the Hilbert transformer Hh(z). ▪ To produce a pair of signals x1(k) and x2(k) that are in phase quadrature, x1(k) must be the output of an allpass filter with a delay of mT/2: where m is even to achieve an integer delay.

Schilling and Harris (2012, p. 370-371)

Quadrature Filters: Hilbert Transformer

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▪ The steady-state values of x1(k) and x2(k) are ▪ x2(k) is a phase-shifted version of x1(k) if Ah( f )  1.

Schilling and Harris (2012, p. 370-371)

Quadrature Filters: Hilbert Transformer

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▪ The Hilbert transformer can be used to create a complex signal of the form ▪ When the Hilbert transformer amplitude response is ideal, i.e., Ah( f ) = 1, the spectrum of the complex

• utput signal is

▪ Since , we then have depending on the sign of f.

Schilling and Harris (2012, p. 370-371)

Quadrature Filters: Hilbert Transformer

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▪ Thus, the spectrum of y(k) is which is zero over the negative frequency range. ▪ The complex signal y(k) is referred to as a half-band signal and is the discrete equivalent of an analytic signal (a complex signal without negative frequency components). ▪ The complex signal y(k) can be used to reconstruct the

• riginal signal x(k) but only occupies half the
• bandwidth. So, y(k) can be transmitted more efficiently

than x(k) in a communication system.

Schilling and Harris (2012, p. 371)

Quadrature Filters: Hilbert Transformer

63

▪ “Recall the frequency shift property of DTFT that if a signal is modulated by a complex exponential, this shifts its spectrum.” ▪ Then, several half-band signals can be translated to different regions of the spectrum and then transmitted simultaneously. ▪ This technique is called frequency-division multiplexing.

Schilling and Harris (2012, p. 371)

Digital Oscillator

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▪ “A discrete-time system that generates a phase quadrature pair at a fixed frequency F0 is called a digital oscillator.” ▪ “A sinusoidal oscillator of frequency F0 will produce the quadrature pair:” ▪ x1(k) can be rewritten as ▪ Similarly,

Schilling and Harris (2012, p. 372)

Digital Oscillator

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▪ This leads to the 2D linear system that has no input. Evaluating and at k = 0 yields the initial conditions x1(0) = 1 and x2(0) = 0. ▪ This system can then be rewritten as where x = [x1, x2]T. ▪ The rotation matrix C rotates x(k-1) counterclockwise about the origin by angle  starting from x(0) along the unit circle.

Schilling and Harris (2012, p. 372)

Digital Oscillator

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▪ “Once a quadrature pair is generated by a digital

• scillator, it can be used to synthesize a more general

periodic signal by post processing x1(k) and x2(k) to generate harmonics.” ▪ The Chebyshev polynomials of the first kind can be

• btained from the recurrence relation

▪ The Chebyshev polynomials of the first kind has the property

Schilling and Harris (2012, p. 372)

Digital Oscillator

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▪ “Since x1(k) is a cosine, any even periodic function can be generated from x1(k) using a suitable linear combination of the Chebyshev polynomials of the first kind.” ▪ “To generate an odd periodic function, we can use the Chebyshev polynomials of the second kind obtained from the recurrence relation with property

Schilling and Harris (2012, p. 372-373)

Digital Oscillator

68

▪ “To avoid aliasing, the ith harmonic must be below the Nyquist (folding) frequency fs/2. Thus, the number of harmonics that can be generated without aliasing is” ▪ “A general periodic signal with period T0 = 1/F0 can be approximated with r harmonics using the truncated Fourier series”

Schilling and Harris (2012, p. 373)

Digital Oscillator

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Given the quadrature pair and the harmonic generation properties the periodic signal y(k) can be rewritten as where

Schilling and Harris (2012, p. 373)

Digital Oscillator: Example

70

Suppose fs = 1 kHz and F0 = 25 Hz. Up to r = 19 harmonics can be generated without aliasing. Suppose and N = 200 points are computed.

Schilling and Harris (2012, p. 373)

The closed curve of y(k) versus x1(k) shows that y(k) is periodic with period F0 = 25 Hz.

Notch Filters

71

▪ Notch filters are filters whose stopband consists of a single frequency. ▪ A unit pulse 1( f ) is a continuous-frequency unit impulse defined as ▪ A notch filter with a notch at F0 is a filter whose magnitude response is ▪ Thus, a notch filter passes all frequency components except F0. Notch filters can be approximated by low-

• rder IIR filters.

Schilling and Harris (2012, p. 374)

Notch Filters

72

▪ An example of notch filters is a filter that blocks DC, i.e., a filter with a notch at F0 = 0 Hz. ▪ An exact DC-blocking filter can be easily implemented using a noncausal filter whose output y(k) is where x is the mean of the input x(k). ▪ For a causal realization of a notch filter, consider a first-

• rder IIR filter with transfer function

▪ Along the unit circle, z = exp( j2 f T), so DC (f = 0 Hz) corresponds to z = 1.

Schilling and Harris (2012, p. 375)

Notch Filters

73

▪ “Thus, the zero of HDC(z) at z = 1 ensures that the DC component of x(k) will be blocked.” ▪ “To get a passband gain of one, we can set HDC(-1) = 1 where z = -1 corresponds to the Nyquist frequency.” ▪ Setting HDC(-1) = 1 and solve for c yields . ▪ Since , this implies that c  1. ▪ Placing the pole at z = r very close to the zero at z = 1 ensures that Anotch( f )  1 for f  F0 which is required to make the notch filter effective:

Schilling and Harris (2012, p. 375)

Notch Filters

74

▪ The 3dB cutoff frequency Fc is the frequency at which ▪ “For a DC notch filter, . Thus, , which means cos(2FcT)  1 and sin(2FcT)  2FcT.” ▪ Then, the square of the magnitude response at f = Fc is ▪ Setting and solving for r yields ▪ The bandwidth of the stopband or notch is F = 2Fc.

Schilling and Harris (2012, p. 375)

Example: DC Notch Filter

75

▪ Consider 3 filters with notch bandwidths of F = [0.4, 0.2, 0.1] Hz with fs = 100 Hz. ▪ Using the formulas: the corresponding poles and gains are

Schilling and Harris (2012, p. 375)

The pole and zero are close together, so their phase contributions almost cancel except near DC. So, the phase response of HDC(z) is close to zero in passband.

Notch Filter

76

▪ For a notch filter with a notch at frequency F0  0, there must be a zero on the unit circle at where . There will be a complex-conjugate pair of zeros at and a matching complex- conjugate pair of poles at ▪ Thus, the general form of a notch filter is ▪ DC notch filter:

Schilling and Harris (2012, p. 376)

Notch Filter

77

▪ A notch filter with notches at F0 and F1 can be realized by using a series or cascade configuration. ▪ If multiple notch frequencies are equally spaced to include a fundamental frequency F0 and its harmonics, then we obtain an inverse comb filter.

Schilling and Harris (2012, p. 376-377)

Resonator

78

▪ Ideal notch filter is designed to stop a single frequency. ▪ Ideal resonator is designed to pass a single frequency called the resonant frequency. ▪ The magnitude response specification of a resonator with resonant frequency F0 is ▪ A notch filter with notch frequency F0 and a resonator with resonant frequency F0 form a power- complementary pair satisfying ▪ This implies that

Schilling and Harris (2012, p. 376-377)

Resonator

79

▪ So, one way to design a resonator is ▪ Applying this to the DC notch filter HDC(z), we obtain the DC resonator ▪ For a general resonator with resonant frequency F0, we need to put poles at to create the resonance, and zeros at z = 1 to ensure a bandpass characteristics.

Schilling and Harris (2012, p. 376-377)

Resonator

80

▪ Resonator with resonant frequencies F0 and F1 can be realized using a parallel configuration ▪ A resonator with multiple resonant frequencies equally spaced to include a fundamental F0 and its harmonics is called a comb filter.

Schilling and Harris (2012, p. 377-378)

Power-Complementary Pair: Example

81

Schilling and Harris (2012, p. 378)

▪ Schilling, R. J. and S. L. Harris, 2012, Fundamentals

• f Digital Signal Processing using MATLAB, Second

Edition, Cengage Learning.

Reference