In this article, we will learn about various filter responses. A filter response helps you understand the difference between the input signal and the output signal of a filter. Keeping an eye on filter responses is necessary to analyze their behavior. To have a better understanding of the processed signal, it is important to analyze this signal and the resulting changes produced by the system.
In electronics, signal analysis is done by understanding the behavior of the signal in the time and frequency domain. In this article, we will look at the important signal responses produced by the two domains: Time and Frequency.
Response in the Time Domain
A disturbance or sudden change in an input signal from its steady state is called a transient. There are methods to study the time domain and frequency domain response of the hardware filter. The time domain response of the hardware filter is analyzed under transient conditions.
Figure 1: Time domain vs. Frequency domain
Two techniques are used to move between the time domain and the frequency domain filter response – Fourier Transform and Laplace Transform. To apply these two techniques, a mathematical function is generated that theoretically models the output of any system across all possible inputs. This is called the transfer function, which is the ratio of the output to the input response time. The impulse response of a filter can be used to define its bandwidth. Time domain response is a practical consideration in communication systems, where different modulation schemes utilize amplitude and phase information.
- impulse response
An infinitely narrow pulse with an infinitely high unit area (small area of high amplitude) is defined as an impulse response. Physically it is impossible to perceive the impulse response. If the filter has an impulse width smaller than the filter rise time (the filter starts filtering), the filter response will be the actual impulse response.
The impulse response of the filter in the time domain is proportional to the bandwidth of the filter in the frequency domain. The narrower thrust means the wider the bandwidth of a filter. The pulse amplitude can be written as ù c /ð, which is proportional to the filter bandwidth. Height increases with wider bandwidths. The pulse width can be written as 2ð/ù c, which is inversely proportional to the bandwidth of a filter. The product of amplitude and bandwidth becomes a constant.
It is not easy to calculate the filter response without using Fourier and Laplace transforms. The Laplace transform converts multiplication into addition and division into subtraction, transforming them into simple, easier-to-handle algebraic equations. The Fourier transform works in the opposite direction of the Laplace transform.
As stated, the impulse response is directly related to the filter bandwidth. Therefore, amplitude discrimination (ability to distinguish between the desired signal and noise) and time are inversely proportional. This is why it is said that filters with the best amplitude response have the worst time response. In technical design, there are types of filters like Butterworth, Chebyshev and Bessel filters. Each filter has its own unique design features. The Chebyshev filter provides better amplitude discrimination than Butterworth, and Butterworth provides better amplitude discrimination than the Bessel filter. Bessel filters are better in the time domain. The time domain classification can be given as: Bessel followed by Butterworth and then Chebyshev.
Increasing the filter order increases the impulse response but results in greater bandwidth limitation, degrading the response time. Degrading the response time means increasing the frequency discrimination and quality factor of the individual section, which implies a longer response time.
- Step response
The integral of the impulse response of a filter is called the step response. The step response is useful in time domain response because it contains the information of a signal in a recognizable view. Most of the generalities applied to the impulse response can be used for the step response. The rise time slope of the step response is equal to the peak response of the impulse response, and the product of rise time and bandwidth is constant. The impulse response has a function of unity, just as the step response also has 1/s. Both expressions are dimensionless and therefore can be normalized.
The step response of a filter is used to determine the envelope distortion (variations in the rate of phase change across frequency) of a modulated signal. Overshoot (when a signal crosses its limited area) and ringing are the two most important parameters of a filter's step response. In an excellent pulse response, overshoot should be minimal. The ringing should subside as quickly as possible so as not to disturb subsequent pulses.
Fig. 2: Overshoot and signal ringing
Real-life communication signals are not made of step or impulse responses, so it is not possible to obtain a complete and accurate output estimate using transient response curves. There are several CAD (Computer Aided Design) software programs that can perform mathematical calculations of impulse and step response.
Frequency Domain Response
Frequency domain response is the quantitative measurement of the phase and amplitude of the output as a function of the input frequency.
There are transfer functions that can satisfy the attenuation and phase requirements of a filter. The transfer function can determine the importance of FDR (frequency domain response) versus TDR (time domain response).
- Butterworth filter response
A Butterworth filter has the smoothest frequency response in the filter's passband. It also has a very simple transfer function equation. It is relatively simple to calculate the coefficient of polynomials, due to the easy transfer function equation.
Figure 3: Butterworth filter response
The best reconciliation between phase response and attenuation is the Butterworth filter. It has no ripple in the stopband and passband, which is why it is sometimes called a maximally flat filter. A Butterworth filter achieves its flatness by wide transition from the passband to the endband, with calculated transient characteristics.
In the S-plane, the normalized pole of the Butterworth filter is on the unit circle. And the pole positions are:
-sin ( (2k-1) ð / 2n ) + j cos ( (2k-1) ð / 2n )k=1.2….n
Where k – number of pole pair, n – number of poles
In the unit circle, the poles are spaced equidistant, meaning there are equal angles between the poles.
u 0 , and Q can be calculated from the given pole locations. The values of the components can then be determined by the values of a filter. Frequency and impedance normalized filters used for passive filters are normalized to an impedance of 1 Ω and frequency of 1 rad/s. This allows comparison of the frequency domain and time domain response of the filters on an equal footing. The normalization of the Butterworth filter is the -3dB response at ù 0 =1.
Butterworth filter element values are more practical and less critical than other filters.
- Chebyshev filter response
A Chebyshev filter has a very sharp transition from the passband to the endband of a filter. Depending on the type of Chebyshev filter used, this sharp transition causes ripples in the passband and endband.
Figure 4: Chebyshev filter response
This filter has a smaller transition region than the Butterworth filter for the filter of the same order, but it has ripples in the passband. The Chebyshev criterion has maximum ripples in the passband, and the Chebyshev filter minimizes the height of the maximum ripple of the Chebyshev criterion.
At DC, these filters have a relative attenuation of 0 dB. The number of ripple cycles in the passband is equal to the filter order. This extends from 0 to the maximum odd-order ripple value; even order filters have a gain equal to the ripple of the passband.
Moving the poles of the Butterworth filter (forming an ellipse) can determine the poles of the Chebyshev filter. This can be done by multiplying the imaginary part by k 1 and the real part of the pole by k R .
And these values can be determined by:
K R = sinh A
K 1 = cosh A
Where:
A = (1/n)sinh -1 (1/Ꜫ)
where n-order of the filter and
Ꜫ = square(10 R -1)
Where:
R = R dB /10
Where:
R dB – ripple of the passband
The 3 dB bandwidth of the Chebyshev filter is given by:
At 3dB = (1/n) cosh -1 (1/Ꜫ)
- Bessel filter response
The Bessel filter has a linear phase response in the passband. Due to this linearity, all signal frequencies are delayed by the same amount of time, making it ideal for image processing and control applications. This filter needs to be higher order to obtain the same transition from passband to stopband as the Chebyshev or Butterworth.
The figure below shows the comparison between the responses of the three filters.
Figure 5: Comparison between Butterworth, Chebyshev and Bessel filters.
A Butterworth filter has good transient behavior with fairly good amplitude, while Chebyshev filters improve amplitude response by sacrificing transient behavior. But the best transient response is optimized by the Bessel filter due to the linear phase in the passband – this means worse frequency response.
The poles of the Bessel filter are on a circle like the Butterworth filter, but are spaced at approximately equal distances, unlike those related to the real axis, rather than equal angular distances. The real and imaginary location of the Bessel filter poles is shown in the figure below.
Fig. 6: Real and imaginary pole location for Bessel filter
To imply the response in the time domain and the frequency domain or to switch between them, a mathematical filter equation is generated, which is done by the transfer function. Therefore, it is necessary to understand some basic functions and properties of a transfer function.
