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DSP [Digital Signal Processing]
A.H.M. Asadul Huq, Ph.D.
[email protected]://asadul.drivehq.com/students.htm
April 13, 2023 A.H. 1
[Design of FIR: Window method]Vol-6
Introduction to FIR Filter• What does "FIR" mean?
- "FIR" means "Finite Impulse Response". • Why is the impulse response "finite" ? – because there is no feedback in the FIR.
- A lack of feedback guarantees that the impulse response will be finite.
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Advantages of FIR (compared to IIR) Filters
• Easily be designed to be "linear phase"– Put simply, linear-phase filters delay the input signal but
don’t distort its phase.• Simple to implement.– On most DSP microprocessors, the FIR calculation can be
done by looping a single instruction.• They are suited to multi-rate applications.• Have desirable numeric properties.
- FIR filters without feedback can usually be implemented using fewer bits.
• Unconditionally stable unlike IIR
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FIR equations
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h(k), k = 0, 1, … , N - 1 are the Impulse Response (coefficients).H(z) is the Transfer Function, N is the Filter Length (No. of Filter Coefficients) of the filter.
Difference Equation
System function (Transfer function) equation
Linear Phase Response
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•"Linear Phase" refers to the condition where the phase response of the filter is a linear (straight-line) function of frequency (excluding phase wraps at +/- 180 degrees). •This results in the delay through the filter being the same at all frequencies. •Therefore, the filter does not cause "phase distortion" or "delay distortion". •The lack of phase/delay distortion is an advantage of FIR filters over IIR and analogue filters in certain systems, for example, in digital data modems.
Phase Delay and Group delay
•The Phase Delay (Tp) or Group Delay (Tg) of the filter is a measure of how a filter modifies the phase characteristics of the signal. •Consider a signal consisting of several frequency components, like: Speech waveform, or a Modulated signal• Phase Delay (Tp) =
is the amount of time delay each frequency component of the signal suffers in going through the filter.
• Group Delay (Tg) = is the average time delay the composite signal suffers at each frequency.
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Problems of Non-linear Phase Filter• A non-linear phase filter will cause phase distortion in the signal that
passes through it.• This is due to the fact that the frequency components in the signal will
each be delayed by an amount NOT proportional to frequency, hence altering their harmonic relationship.
• This distortion is extremely undesirable in applications involving: • Music, Data transmission, Video, and Bio-medical applications • This distortion can be avoided by using filters with linear phase
characteristics.• Non-linear phase filters will distort the audio of AM broadcast signals, blur
the edges of television video images, blunt the sharp edges of received radar pulses, and increase data errors in digital communication signals.
• None-linear phase filters are also called “minimum phase”
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Types of linear phase FIR filters
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4 types of linear phase FIR filters
Linear-Phase FIR types
• Type 1 –most versatile• Type 2 – frequency response is always 0 at
ω=π. (not suitable as a high-pass filter)• Type 3 and 4 – Introduce a π/2 phase shift,
frequency response is always 0 at ω=π. (not suitable as a high-pass filter)
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Type 1 and 2 FIR coefficients• Type-1: Positive symmetry, Odd length of Coefficients• Example impulse response -
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• Type-2: Positive symmetry, Even length of Coefficients• Example impulse response -
Type 3 and 4 Fir CoefficientsType 3 [Negative Symmetry, Odd Length (Coefficients)]: 90° phase shift Always zero at f = 0, hence unsuitable as a Low-pass filter. In addition, always zero at f = 0.5, hence unsuitable as a High-pass filter. The sample impulse response is given by:
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Type 4 [Negative Symmetry, Even Length (Coefficients)]: 90° phase shift Always zero at f = 0, hence unsuitable as a Low-pass filter. The example impulse response is given by:
Steps for designing FIR Filter
5 Steps:1. Determine the filter specifications2. Choose a suitable filter structure3. Calculate the filter coefficient values4. Analysis the finite wordlength effect5. Construct the filter using software and/or
hardware.
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FIR filter design flowchart
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Ideal Specifications
• Unity magnitude over passband• Zero magnitude over stopband• Linear response over passband
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1
0 c –c
HLP(e j)
0 c –c
1
HHP (e j)
11–
–c1 c1 –c2 c2
HBP (e j)
1
–c1 c1 –c2 c2
HBS(e j)
Tolerance scheme: magnitude-frequency response of a LP Filter
The following parameters are of interest:
• δp: Passband Deviation
• δs: Stopband Deviation
• fp or ωp : Passband Edge Frequency
• fs or ωs Stopband Edge Frequency
• Fs: Sampling Frequency
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Tolerance scheme with normalized filter parameters
• passband= 0.18 – 0.33• transition band= 0.14 – 0.18
and 0.33 – 0.37• stopband= 0 – 0.14 and 0.37
– 0.5• stopband deviation, δs= 0.001
• passband deviation, δs= 1.05, 1, 0.95
• 1 + δp= 1.05
• 1 – δp = 0.95
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Un-normalized filter parameters Normalized filter parameters• passband= 0.18 – 0.33• transition band= 0.14 – 0.18 and
0.33 – 0.37• stopband= 0 – 0.14 and 0.37 –
0.5• stopband deviation, δs= 0.001
• passband deviation, δs= 1.05, 1, 0.95
• 1 + δp= 1.05
• 1 – δp = 0.95
Un-normalized filter parametersWith sampling frequency of 10 kHz: • Passband=1.8 - 3.3 kHz• Stopband(s)=0 - 1.4 kHz and 3.7 -
5 kHz• Stopband Attenuation (As)=-20
log10 δs
• =-20 log10 (0.001)• =60 dB• Passband Ripple (Ap)=20 log10 δp
• =20 log10 (1 + 0.05)• =0.42 dB
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Normalized filter frequencies are also expressed as:ω=2.fn π rad/sec (specially for matlab)
Coefficient calculation MethodsWindow Method: • Offers a very simple and flexible way of computing FIR filter coefficients.• However, does not allow the over the filter parameters. designer adequate
control
The Optimal Method • With efficient and easy-to-use programs it is now widely used in industry.• For most applications, this method will yield the desired FIR filter.• Should be the first choice, unless a particular application dictates otherwise, or a
CAD facility is unavailable. The Frequency Sampling Method • The main attraction is that it allows for a recursive realization of FIR filters, which
can be computationally very efficient. • However, it also lacks flexibility in specifying or controlling filter parameters. •
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Steps of Window method
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Step 1: Specify the “ideal” or desired frequency response of the filter, HD(ω). Step 2: Obtain the IDEAL impulse response, hD(n), of the desired filter by evaluating the IDFT. Step 3: Select a window function w(n) that satisfies the passband and stopband attenuation specifications. Determine the filter length N. Step 4: Multiply the ideal coefficients by the selected window function to get the filter coefficients h(n) = hD(n) . w(n)
Ideal Impulse response for LP Filter
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Note that hD(n) is symmetrical about n = 0 (i.e. hD(n) = hD(-n)), so the filter will have linear phase response. Although hD(n) decreases as we move away from n = 0, the impulse
response is infinite in length (as n = ±∞).Hence, this is not a FIR filter!
Ideal frequency response for LP
Ideal (infinite)Impulse response for LP
Ideal Impulse responses for different filter types
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Truncation of Impulse response
• A possible solution is to truncate the ideal impulse response by setting hD(n) = 0 for n greater than M (equivalent to a rectangular window).
• Rectangular window function - w(n)=1, where, n = 0, 1, …, ± (N - 1)/2 = 0, elsewhere• A practical approach is to multiply the ideal impulse response, hD(n) by a
suitable window function, w (n), whose duration is finite. • This way the resulting impulse response decays smoothly towards zero. Finally, • The ideal impulse response, hD(n) is multiplied by finite duration window
w(n), to obtain the filter coefficients, h(n).
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Truncation Effect on frequency response
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13 coefficients 25 coefficients
Infinite coefficients (no truncation)
Ideal impulse response hD(n)is multiplied by finite duration window w(n) to get filter coefficients h(n)
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)()()( nwnhnh D
Types of Window functions
• Rectangular• Hamming• Hanning• Blackman• Kaiser
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Features of popular window functions
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Window functions in time domain
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Different window functions for N=61
0 20 40 600
0.2
0.4
0.6
0.8
1
1.2Blackman Window
0 20 40 600
0.2
0.4
0.6
0.8
1
1.2Rectangular Window
0 20 40 600
0.2
0.4
0.6
0.8
1
1.2Hamming Window
0 0.2 0.4 0.6 0.8-150
-100
-50
0
Normalized Frequency
|W(f
)| (d
B)
|W(f)| in dB for Blackman Window w(n)
0 0.2 0.4 0.6 0.8-100
-80
-60
-40
-20
0
Normalized Frequency
|W(f
)| dB
|W(f)| for Hamming window w(n)
0 0.2 0.4 0.6 0.8 1-100
-80
-60
-40
-20
0
Normalized Frequency ( rad/sample)
|W(f
)||W(f)| (dB) vs f for rectanular window w(n)
Window functions in Frequency domain
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(a) Rectangular , (b) Hamming, (c) Blackman windows for N = 61.
Rectangular Hamming
Blackman
Example
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Calculate the coefficients of an FIR lowpass filter to meet the specifications given below using the WINDOW method.[E.C. Ifeachor 2/e P. 359]
Passband cutoff frequency = 1.5 kHzTransition width = 0.5 kHz
Stopband attenuation > 50dBSampling frequency = 8 kHz
Solution: See appendix
DSP Lecture
FIR Filter and Window Method
THE ENDTHANK YOU
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