Question

Consider the design of a low-pass filter meeting the following requirements:

i)The pass-band attenuation is less than 0.1 dB for frequencies less than 5 k rad. /sec. and is less than 0.5 dB for frequencies less than 20 k rad. /sec. ii) The stop-band loss is at least 50 dB for frequencies greater than 100 k rad/sec.
iii) The 3 dB cutoff frequency is less or equal 30 k rad. /sec.
a) Find the order of a Butterworth filter to meet these requirements.

Answer 1

To find the order of a Butterworth filter that meets the given requirements, we need to consider the number of poles in the filter. The minimum order of the Butterworth filter needed to meet the given requirements is 1.

Step-by-step explanation:

To find the order of a Butterworth filter that meets the given requirements, we need to consider the number of poles in the filter. The number of poles in a Butterworth filter is equal to the order of the filter. In this case, the pass-band attenuation is less than 0.1 dB for frequencies less than 5 kHz and less than 0.5 dB for frequencies less than 20 kHz. The 3 dB cutoff frequency is less than or equal to 30 kHz. To meet these requirements, we need to find the minimum order of the Butterworth filter that will provide the desired attenuation at the desired frequencies.

Based on the given requirements:

• For frequencies less than 5 kHz, the required attenuation is less than 0.1 dB. This corresponds to a gain of approximately 1.002. Considering the Butterworth filter response, the gain at 5 kHz will be approximately -3 dB or 0.707. This means that the gain at 5 kHz is already higher than the required attenuation. Therefore, we can ignore this frequency in determining the order of the filter.
• For frequencies less than 20 kHz, the required attenuation is less than 0.5 dB. This corresponds to a gain of approximately 1.005. Considering the Butterworth filter response, the gain at 20 kHz will be approximately -0.6 dB or 0.88. This means that the gain at 20 kHz is already higher than the required attenuation. Therefore, we can ignore this frequency as well in determining the order of the filter.
• The 3 dB cutoff frequency is less than or equal to 30 kHz. The -3 dB cutoff frequency is the frequency at which the gain of the filter drops to 0.707. Based on the Butterworth filter response, this occurs at the -3 dB cutoff frequency, which is the same as the 3 dB cutoff frequency. Therefore, we can use the -3 dB cutoff frequency as the maximum frequency to consider.

With these considerations, we can conclude that the minimum order of the Butterworth filter needed to meet the given requirements is 1.