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[SOWHAT0819]

Design a first-order low-pass digital Chebyshev filter with a cut-off frequency of 3.8kHz and 0.5 dB ripple on the pass-band using a sampling frequency of 15,000Hz.

Step 1

Given data,

Chebyshev low pass filter,

Passband ripple, n = 0.5dB
cutoff frequency, fcut = 3.8kHz
Sampling frequency, fs = 15000Hz

Now firstly we have to find wd,
wd = 2 x π x fcut = 2 × π x 3,800= 7,600π rad/s

now wo;

wo = 2 x fs x tan(wd/2fs)= 2 x 15,000 x tan (7,600π/2 x 15,000) = 416.74 rad/s

step 2
therfore we can say that,

Hp(s) = 1.9652/ s+1.9652

now we can wite,

Hp(s) = (1.9652/(s/416.74)+1.9652 x 416.74) / 416.74

Hp(s) = 8118.977/ s+818.977

= 8118.977/ (416.74 x z-1/z+1)+818.977

= 8118.977 (z+1)/ 1,235.717z + 402.237

= 2.036(z+1)/ 3.072 +1

Answer

Hence finally we can say that,

These filters are used to distinguish one frequency band from another. and we get final response as,
Hp(s)= 2.036(z+1)/ 3.072z +1


With this information use this to solve this:
Design a first-order low-pass digital Chebyshev filter with a cut-off frequency of 2.5kHz and 0.5 dB ripple on the pass-band using a sampling frequency of 14,500Hz.

 

[🤖]

To design a first-order low-pass digital Chebyshev filter, we need to follow a few steps:

Step 1: Given data

• Passband ripple: 0.5 dB
• Cut-off frequency: 2.5 kHz
• Sampling frequency: 14,500 Hz

Step 2: Calculate wd (angular cut-off frequency)
wd = 2 * π * fc = 2 * π * 2,500 = 15,707 rad/s

Step 3: Calculate wo (normalized cut-off frequency)
wo = 2 * π * fs * tan(wd / (2 * fs)) = 2 * π * 14,500 * tan(15,707 / (2 * 14,500)) = 2.162 rad/s

Step 4: Determine the transfer function
H(s) = 1 / (s/wo + 1)

Step 5: Convert the transfer function to digital form
Using the bilinear transformation, we can convert the transfer function from s-domain to z-domain with the substitution s = 2 * (z - 1) / (T * (z + 1)), where T is the sampling period.

H(z) = H(s) |s = 2 * (z - 1) / (T * (z + 1))

H(z) = 1 / (2 * (z - 1) / (T * (z + 1)) / wo + 1)

Simplifying further, we get:

H(z) = (T * wo) / (2 * (z + 1) + T * wo * (z - 1))

Step 6: Normalize the coefficients
To normalize the coefficients, divide the entire transfer function by the coefficient of z^0.

H(z) = (T * wo / 2) / (1 + ((T * wo) / 2) * (z - 1) / (z + 1))

In this case, the normalized coefficients are:
B0 = T * wo / 2 = (1 / 14,500) * 2.162 = 0.000297

A0 = 1

A1 = -(T * wo / 2) = -(1 / 14,500) * 2.162 = -0.000297

Step 7: Final result
The final first-order low-pass digital Chebyshev filter with a cut-off frequency of 2.5 kHz and 0.5 dB ripple on the pass-band using a sampling frequency of 14,500 Hz can be represented by the transfer function:

H(z) = (0.000297) / (1 - 0.000297 * (z - 1) / (z + 1))

Please note that the calculated values are approximate due to rounding.