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Lecture 23 Filters Hung-yi Lee

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Filter Types co : cutoff frequency Lowpass filterHighpass filter Bandpass filter Notch filter Bandwidth u - l

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Real World Ideal filter

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Transfer Function – Rules Filter is characterized by its transfer function The poles should be at the left half of the s-plane. We only consider stable filter. Given a complex pole or zero, its complex conjugate is also pole or zero.

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Transfer Function – Rules Filter is characterized by its transfer function :improper filter As the frequency increase, the output will become infinity. :proper filter We only consider proper filer. The filters consider have more poles than zeros.

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Filter Order The order of the denominator is the order of the filter. Order = n order=1 order=4

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Outline Textbook: Chapter 11.2 Second-order Filter Highpass Filter Lowpass Filter Notch Filter Bandpss Filter First-order Filters Highpass Filter Lowpass Filter

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First-order Filters

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Firsr-order Filters Case 1: 1 pole, 0 zero first order zero or first order 1 pole 0 or 1 zero Case 2: 1 pole, 1 zero

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Firsr-order Filters - Case 1 Pole p is on the negative real axis Magnitude decrease Phase decrease As ω increases Lowpass filter

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Firsr-order Filters - Case 1 Amplitude of the transfer function of the first-order low pass filter Ideal Lowpass filter First-order Lowpass filter

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Find cut-off frequency ω co of the first-order low pass filter Firsr-order Filters - Case 1 Lowpass filter At DC Find cut-off frequency ω co such that

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Firsr-order Filters - Case 2 Case 2-1: Absolute value of zero is smaller than pole Zero can be positive or negative Magnitude is proportional to the length of green line divided by the length of the blue line Low frequency ≈ |z|/|p| The low frequency signal will be attenuated If z=0, the low frequency can be completely block Not a low pass Because |z|<|p|

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Firsr-order Filters - Case 2 Case 2-1: Absolute value of zero is smaller than pole Magnitude is proportional to the length of green line divided by the length of the blue line High frequency The high frequency signal will pass High pass If z=0 (completely block low frequency)

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First-order Filters - Case 2 Find cut-off frequency ω co of the first-order high pass filter (the same as low pass filter)

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First-order Filters - Case 2 Case 2-2: Absolute value of zero is larger than pole Low frequency ≈ |z|/|p| The low frequency signal will be enhanced. Neither high pass nor low pass Because |z|>|p| High frequency: magnitude is 1 The high frequency signal will pass.

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First-order Filters Consider v in as input If v l is output If v h is output Lowpass filter Highpass filter (pole)

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First-order Filters (pole)

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Cascading Two Lowpass Filters

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The first low pass filter is influenced by the second low pass filter!

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Cascading Two Lowpass Filters

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Second-order Filters

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Second-order Filter Case 1: Case 2: Case 3: Must having two poles No zeros One zeros Two zeros Second order 2 poles 0, 1 or 2 zeros

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Second-order Filter – Case 1 Case 1-1Case 1-2

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Second-order Filter – Case 1 As ω increases The magnitude monotonically decreases. Decrease faster than first order low pass Case 1-1 Real Poles The magnitude is

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Second-order Filter – Case 1 Case 1-2 Complex Poles As ω increases, 1. Increase The magnitude is l 1 decrease first and then increase. What will happen to magnitude? l 2 always decrease 2. Decrease 3. Increase, then decrease 4. Decrease, then increase

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Second-order Filter – Case 1 Case 1-2 Complex Poles 1. Increase What will happen to magnitude? 2. Decrease 3. Increase, then decrease 4. Decrease, then increase If ω > ω d l 1 and l 2 both increase. The magnitude must decrease.

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Second-order Filter – Case 1 Case 1-2 Complex Poles Minimize Maximize the magnitude When ω < ω d

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Second-order Filter – Case 1 Minimize (maximize)

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Second-order Filter – Case 1 Lead to maximum The maxima exists when Peaking No PeakingPeaking

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Second-order Filter – Case 1 Lead to maximum The maxima exists when Peaking Assume

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Second-order Filter – Case 1 For complex poles

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Second-order Filter – Case 1 Q times of DC gain Q times

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Second-order Filter – Case 1 Lead to maximum For complex poles

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Second-order Filter – Case 1 Lead to maximum The maximum exist when The maximum value is

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Second-order Filter – Case 1 (No Peaking) Case 1-2 Complex Poles Case 1-1 Real Poles Which one is considered as closer to ideal low pass filter?

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Complex poles Peaking (Butterworth filter)

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Butterworth – Cut-off Frequency ω 0 is the cut-off frequency for the second-order lowpass butterworth filter (Go to the next lecture first)

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Second-order Filter – Case 2 Case 2: 2 poles and 1 zero Case 2-1: 2 real poles and 1 zero

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Second-order Filter – Case 2 Case 2: 2 poles and 1 zero Case 2-1: 2 real poles and 1 zero Bandpass Filter flat

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Case 2-2: 2 complex poles and 1 zero Second-order Filter – Case 2 Zero Two Complex Poles + -40dB +20dB

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Case 2-2: 2 complex poles and 1 zero Second-order Filter – Case 2 Zero Two Complex Poles + -40dB +20dB -40dB -20dB +20dB

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Case 2-2: 2 complex poles and 1 zero Second-order Filter – Case 2 Zero Two Complex Poles + -40dB +20dB -20dB +20dB Bandpass Filter Highly Selective

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Bandpass Filter Bandpass filter: 2 poles and zero at original point bandpass filter Bandwidth B = ω r - ω l B

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Bandpass Filter Bandpass filter: 2 poles and zero at original point Find the frequency for the maximum amplitude

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Bandpass Filter Transfer function of bandpass filter is maximized when ω 0 is center frequency

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Bandpass Filter - Bandwidth B Four answers? Pick the two positive ones as ω l or ω r

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Bandpass Filter - Bandwidth B Q is called quality factor Q measure the narrowness of the pass band

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Second-order Filter – Case 3

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Thank you!

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Higher order filter Buttorworth Notch filter for humming Different kinds of filter: active, passive ……

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Radio Amplifier P1562

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Suppose this band-stop filter were to suddenly start acting as a high- pass filter. Identify a single component failure that could cause this problem to occur: If resistor R 3 failed open, it would cause this problem. However, this is not the only failure that could cause the same type of problem!

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Giutar capacitor https://www.youtube.com/watch?v=3I62Xfhts 9k

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Algorithmic implementation wiki

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High pass They are used as part of an audio crossover to direct high frequencies to a tweeter while attenuating bass signals which could interfere with, or damage, the speaker. When such a filter is built into a loudspeaker cabinet it is normally a passive filter that also includes a low-pass filter for the woofer and so often employs both a capacitor and inductor (although very simple high-pass filters for tweeters can consist of a series capacitor and nothing else).audio crossovertweeterloudspeakerpassive filterlow-pass filterwooferinductor

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64 Any second-order bandpass filter may be described by Where quality factor: the network is underdamped when ½ : damping coefficient

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65 The transfer function of a second-order notch filter is The notch effect comes from the quadratic numerator The notch width is B = O / Q

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66 Table 11.3 Simple Filter Type Transfer Function Properties Lowpass Highpass Bandpass Notch

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Example 11.6 Design of a Bandpass Filter 67 bandpass filter: L = 1 mH, R w = 1.2 , C = ?, R = ? frequency: 20kHz ± 250Hz

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From Wiki Butterworth filter – maximally flat in passband and stopband for the given order Butterworth filter Chebyshev filter (Type I) – maximally flat in stopband, sharper cutoff than Butterworth of same order Chebyshev filter (Type I) Chebyshev filter (Type II) – maximally flat in passband, sharper cutoff than Butterworth of same order Chebyshev filter (Type II) Bessel filter – best pulse response for a given order because it has no group delay ripple Bessel filter Elliptic filter – sharpest cutoff (narrowest transition between pass band and stop band) for the given order Elliptic filter Gaussian filter – minimum group delay; gives no overshoot to a step function. Gaussian filter

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Filter A filter is a circuit that is designed to pass signals with desired frequencies and reject the others. Magnitude ratio Filter Only input signal at these frequencies can pass

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Loudspeaker for home usage with three types of dynamic drivers 1. Mid-range driver 2. Tweeter 3. Woofers

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Second Order Lowpass Filter Compare with 1 st order low pass filter As the frequency increases, the amplitude ratio drops faster than 1 st order low pass filter

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Firsr-order Filters - Case 1 |p| is the cut-off frequency Larger cut-off frequency Smaller cut-off frequency Lowpass filter

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Firsr-order Transfer Function - Case 2 Highpass? Lowpass? Both possible? Absolute value of pole is equal to zero Positive zero can cause phase shift Magnitude is the length of green line divided by the length of the blue line All pass filter Phase is the angle of the green minus the negative one

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First Order Lowpass Filter Maximum: Cut-off Frequency ω co :

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Cut off frequency

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60Hz Hum

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First-order Filters Are there anything wrong? The highpass and lowpass filters have the same cut- off frequency.

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