ELEN 5346/4304 DSP and Filter Design Fall Lecture 8: LTI filter types Instructor: Dr. Gleb V. Tcheslavski Contact: Office Hours: Room 2030 Class web site: p/index.htm p/index.htm

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI IIR filters Note: we are interested in BIBO: |H(  )|,  H(  ) 1. First-order Lowpass (8.2.1) (8.2.2)(8.2.3) Monotonically decreases with  It is typical to have a maximum magnitude of 1, i.e. a gain of 0 dB. Therefore: Where for stability: |  | < 1 (8.2.4)

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI IIR filters Therefore: (8.3.1)  = 0.7 zplane A first-order system can also be expressed as: (8.3.2) Apparently, here: (8.3.3) (8.3.4)

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI IIR filters A square magnitude function can be evaluated as a square of (8.3.1): Its derivative with respect to frequency is always non-positive, proving that the frequency response monotonically decreases. The passband of a LPF is usually defined by the frequency range from 0 to  c, called the 3-dB cutoff frequency. Here the gain of -3 dB is with respect to the gain at  = 0. (8.4.1) (8.4.2)

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI IIR filters To determine the  c, we equate the squared magnitude to ½: The last equation can be solved either for  c or  : (8.5.1) (8.5.2) (8.5.3) (8.5.4)

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI IIR filters 2. First-order Highpass For a maximum magnitude of 1, i.e. a gain of 0 dB: Where for stability: |  | < 1,  c and  can be found by (8.5.3) and (8.5.4) (8.6.1)  = 0.7

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI IIR filters 3. Second-order Bandpass A bandpass filter cannot be obtained by a first-order real-coefficient transfer function. The lowest order transfer function must have a pair of complex conjugate poles and zeroes at z = +1 and z = -1. This transfer function has a pair of complex conjugate poles Here, for stability: |  | < 1 and |  | < 1 (8.7.1) (8.7.2)

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI IIR filters For a maximum magnitude of 1: The center frequency of the IIR BPF can be found as: Or: Where: The 3-dB bandwidth (the difference between 3-dB cutoff frequencies) is: (8.8.1) (8.8.2) (8.8.3) (8.8.4) (8.8.5)

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI IIR filters  = 0.6  = 0.5 (8.9.1) The quality factor:

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI IIR filters 4. Resonator For a maximum gain of 1: (8.10.1) (8.10.2) (8.10.5) (8.10.3) (8.10.4) poles

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI FIR filters (8.11.1) (8.11.2) (8.11.3) where The impulse response: r = 0.9  0 = 0.2 

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI FIR filters 5. Sinusoidal oscillator – a resonator with poles on the uc (8.12.2) (8.11.3) where (8.12.1) The impulse response, assuming is: (8.11.4)  0 = 0.2 

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI IIR filters 6. Notch filter For a maximum magnitude of 1: (8.13.1) Here, for stability: |  | < 1 and |  | < 1 The transfer function has a zero at the notch frequency  0 = cos -1 (  )  = 0.6  = 0.5

ELEN 5346/4304 DSP and Filter Design Fall Group (envelope) delay So far, we discussed a magnitude of frequency response only. It turns out that the phase of system frequency response is of importance too. The derivative of phase (of system’s frequency response) with respect to frequency has units of time and is called a group (envelope) delay: (8.14.1) It is a time delay that a signal component of frequency  undergoes as it passes through the system. When phase is linear, the group delay is a constant; therefore, all signal components are delayed by the same time  no phase distortions – design goal… IIR filters, in general, do not have linear phase! Additionally, we may need to compensate for group delays introduced by other filters; therefore…

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI IIR filters 7. Allpass filter – used in phase equalizers Example: Specifying the polynomial that has roots at z = z 0 Therefore (8.15.1) (8.15.2) (8.15.3) (8.15.4)

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI IIR filters H(z) has poles at which must correspond to a BIBO system. H(z) also has zeros at - reciprocal! In general: (8.16.1) A first-order filter: (8.16.2) (8.16.3) (8.16.4) For stability: r < 1, and the group delay is always non-negative (causal system!)

ELEN 5346/4304 DSP and Filter Design Fall Cascade of filters Note: by cascading simple filters, we can design filters with sharper magnitude response; for example, a cascade of K identical first order LPFs will result in a system with the overall transfer function For stability: (8.17.1) (8.17.2) A frequency response of a single bandpass IIR, a cascade of two, and a cascade of three identical bandpass IIR sections:  = 0.2;  = 0.34.

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI IIR filters 8. Comb filter Starting with a simple filter We form New zeros at New poles at (8.18.1) (8.18.2) (8.18.3) (8.18.4) A comb filter is easy to concentrate on harmonics. We can emphasize or attenuate them.

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI IIR filters z 1 = 0.6 p 1 = -0.5 L = 5

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI FIR filters 1. First-order Lowpass (8.20.1) (8.20.2) 3 dB cutoff frequency: (8.20.3) The phase characteristic of this filter is linear.

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI FIR filters For a cascade of M first-order FIR LPFs, the cutoff frequency will be (8.21.1) Simple FIR filters are inexpensive to implement. Much better approximations of ideal frequency response can be obtained by higher order FIR filters.

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI FIR filters 2. First-order Highpass 3 dB cutoff frequency: (8.22.3) (8.22.1) (8.22.2) A cascade of filters will make frequency characteristic better…

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI FIR filters 3. Notch (8.23.1) phase shift This filter is NOT distortion- less! We will call it a Generalized Linear Phase (GLP) FIR. “don’t care region”

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI FIR filters 4. Moving Average (MA) filter (8.24.1) M = 10 this pole is cancelled by a zero

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI FIR filters 5. Comb Comb filters can be generated from LP prototypes: From HP prototypes: or from more complicated prototype filters, such as MA: (8.25.1) (8.25.2) (8.25.3)

ELEN 5346/4304 DSP and Filter Design Fall Types of LTI FIR filters L = 5