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LABORATORY WORK № 7
ANALOG FILTERT DESIGN
BRIEF THEORETICAL INFORMATION
A filter is one, which rejects unwanted frequencies from the input signal and allows the desired frequencies to obtain the required shape of output signal.
The range of frequencies of signal that are passed through the filter is called passband and those frequencies that are blocked is called stopband.
The filters are of different types:
1. Low-pass filter (LPF). 2. High-pass filter (HPF).
3. Band-pass filter (BPF). 4. Band-stop (reject) filter (BSF).
An ideal filter will have an amplitude response that is unity (or at a fixed gain) for the frequencies of interest (called the pass band) and zero everywhere else (called the stop-band). The frequency at which the response changes from pass-band to stop-band is referred to as the cut-off frequency, . Figure 1(a) shows an idealized low-pass filter. In this filter the low frequencies are in the pass band and the higher frequencies are in the stop band. The PHF passes the high frequencies (they are in the pass band) and all the low frequencies are in the stop band (Figure 1 b – idealized HPF). If a high filter and low –pass filter are cascaded, a band pass filter is created. The BPF passes a band of frequencies berween a lower cut-off frequency, and an upper frequency, . The frequncies below and above are in the stop band. An idealized BPF is shown in Figure 1(c). The band-stop filter passes frequencies below and above . The band from to are in the stop band. Figure 1 (d) shows an idealized BSF.
Figure 1 Idealized Filter Frequency Responses
The idealized filters defined above, unfortunately, cannot be easily built. The transition from pass band to stop band will not be instantaneous, but instead there will be a transition region. Stop band attenuation will not be infinite. Practical filters are usually designed to meet a set of specifications.
To obtain a physically realizable filter, it is necessary to relax some of the requirements of the ideal filters. Figure 2 shows the frequency characteristics of physically realizable versions of various ideal filters. The upper and lower bounds for the gains are indicated by the shaded line, while examples of the frequency characteristics of physically realizable filters that satisfy the specified bounds are shown using bold lines. These filters are referred to as non-ideal or practical filters and are different from the ideal filters in the following two ways:
pass bands ;
stop bands .
The oscillations within the pass and stop bands are referred to as ripples. In Figure 2, the pass band ripples are constrained to a value of for LPF, HPF, and BPF. In the case of BSF, the pass band ripples are limited to and , corresponding to the two pass band. Similarly, the stop band ripples in Figure 2 are constrained to for LPF, HPF and BSF. In the case of BSF, the stop band ripples are limited to and for the two stop bands of the BSF.
CUT-OFF FREQUENCY
An important parameter in the design of continuous time filters is the cut-off frequency of the filter, which is defined as the frequency at which the gain of the filter drops to 0.7071 times its maximum value. Assuming a gain of unity within the pass band, the gain at the cut-off frequency is given by 0.7071 or -3 dB on a logarithmic scale. Since the cut-off frequency lies typically within the transitional band of the filter, for a low pass filter
.
Since the equality implies a transitional band of zero bandwidth, this equality is only valid for ideal filters.
Lowpass and highpass filters usually have the following requirements:
There are four popular types of standard filters:
- Butterworth
- Chebysev Type I
- Chebyshev Type II
- Elliptic.
Figure 2 Frequency Characteristics of Practical Filters
(a) Practical Low Pass Filter; (b) Practical High Pass Filter;
(c) Practical Band Pass Filter; (d) Practical Band Stop Filter
NORMALIZED TRANSFER FUNCTIONS
A normalized low pass transfer function is one in which the pass band edge radian frequency is set to 1 rad/sec. Of course, this seems a rather unusual frequency, since seldom would a low pass filter be required to have such a low frequency. However, the technique actually allows the filter designer considerable latitude in designing filters because a normalized transfer function can easily be unnormalized to any other frequency.
BUTTERWORTH FILTERS
The Butterworth filter is the best compromise between attenuation and phase response. It has no ripple in the pass band or the stop band, and because of this is sometimes called maximally flat filters. The Butterworth filter achieves its flatness at the expense of a relatively wide transition region from pass band to stop band, with average transient characteristics. The low pass Butterworth polynomial has an all-pole transfer function with no finite zeros present.
The normalized Butterworth low pass filter of the n-th order is given by magnitude response function
,
where n is the order of the filter, n=1, 2,…
The normalized poles of the Butterworth filter fall on the unit circle (in the s-plane). The pole positions in left-half plane are defined via following relation
, (1)
where
; k=1, 2,…, n. (2)
Draw your attention, that the poles positions in the right-half plane are defined as follows
,
where are defined by equation (2); k=1, 2,…, n. k is the pole pair number; and n is the number of poles.
BUTTERWORTH DENORMALIZED APPROXIMATION FUNCTIONS
Assume that it is necessary to define a low pass Butterworth filter with cut-off frequency rad/s. In order to obtain such a filter we have to substitute each symbol of normalized low pass filter transfer function by . The resulting low pass filter will possesses with the cut-off frequency equal to . In this case, the low pass Butterworth filter of the n-th order and cut-off frequency is given by following magnitude response
. (3)
The order of the Butterworth filter is dependent on the specifications provided by the user. These specifications include the edge frequencies and gains. The standard formula for the Butterworth order calculation is given by
(n needs to be round to integer value). (4)
CHEBYSHEV FILTERS
There are two types of Chebysheshev filters. Type I Chebyshev filters are all-pole filters that exhibit equiripple behaviour in the pass band and a monotonic characteristic in the stop band. On the other hand, the family type II Chebyshev filters contains both poles and zeros and exhibits a monotonic behaviour in the pass band and equiripple behaviour in the stop band. The zeros of this class of filters lie on the imaginary axis in the s- plane.
The magnitude squared of frequency response characteristic of a type I Chebyshev filter is given as
, (5)
where δ is a parameter of the filter related to the ripple in the pass band and TN(ω) is the N-th order Chebyshev polynomial defined as
, .
The Chebyshev polynomial can be generated via recursive equation
, N=1, 2, … (6)
where and . From equation (6) we obtain
and so on.
Main properties of Chebyshev type I polynomials
1. for all .
2. for all N.
3. All the roots of the polynomials TN(ω) occur in the interval .
The filter parameter δ is related to the ripple in the pass band, as illuctrated in Figure 3, for N odd and N even. For N odd, and hence . On the other hand, for N even and hence . At the cut-off frequency , we have , so that
or, equivalently,
, where δ is the value of the pass band ripple.
For a given set of specifications, it is possible to determine the order of the filter from equation
, (7)
where n needs to be round to integer value.
Figure 3 Type I Chebyshev Filter Characteristics
STANDARD TASK FOR LABORATORY WORK
Table 1
BUTTERWORTH LOW PASS CHARACTERISTICS
|
Variants |
1 |
2 |
3 |
4 |
5 |
6 |
|
ωс, rad/s |
0.25 |
7 |
1·103 |
5 |
0.75 |
0.85 |
|
n, order |
2 |
1 |
1 |
2 |
1 |
2 |
.
Define an order of Butterworth low pass filter that satisfies the following requirements (see eq. 4)
Table 2
CHEBYSHEV FILTER ORDER
|
Variants |
1 |
2 |
3 |
4 |
5 |
6 |
|
n, order |
3 |
8 |
4 |
5 |
7 |
4 |
|
Variants |
7 |
8 |
9 |
10 |
11 |
12 |
|
n, order |
6 |
4 |
5 |
6 |
7 |
3 |
Figure 4 Analog Filter Design
Take the block for analog filter simulation from Signal Processing Blockset\ Filtering\Filter Designs\ Analog Filter Design
Define Main properties for low pass Butterworth filter:
To generate a harmonic signal use Signal Generator block from Simulink\Sources Library. Open Signal Generator block parameters and define the following parameters
Type of the harmonic signal is given below by y.
a) Generate a signal that has the following form
y=K0·2·sinω+noise,
where the sine wave frequency is equal to ω=20 rad/sec. Apply the filtering via Butterworth filter.
b) Generate a signal than involves a low period and high period components, respectively. Hence, the signal y takes the following form
y=K0··(2·sinω1+2· sinω2),
with ω1=20 rad/sec and ω2=80 rad/sec.
Apply the simulation according to individual variants (see Table 3): Butterworth approximation, cut-off frequency ωс; gain K0. Set the simulation time to 5.
Table 3
INITIAL DATA FOR ANALOG FILTER DESIGN
|
Varian |
1 |
2 |
3 |
4 |
5 |
6 |
|
ωс, rad/sec |
30 |
30 |
30 |
30 |
30 |
30 |
|
K0 |
1 |
2 |
3 |
4 |
5 |
6 |