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Chebyshev vs. Butterworth Chebyshev vs. Butterworth Filters: Speed, Quality Factor, and Making the Right Choice Introduction: When it comes to selecting a filter for signal processing, Chebyshev and Butterworth filters are two of the most popular options. Both filters have their unique strengths and weaknesses, and choosing the right one can greatly impact the effectiveness of your signal processing. In this post, we'll explore why the Chebyshev filter is faster than the Butterworth filter and delve into the trade-offs associated with the quality factor of the Chebyshev filter. We'll also provide an explanation of the quality factor to help you make an informed decision. Quality Factor: A Brief Overview The quality factor, also known as the Q-factor, is a dimensionless parameter that represents the "sharpness" of a filter's frequency response. In other words, it measures how well a filter can separate signals with close frequencies. A higher Q-factor indicates a more selective filter, with a steeper roll-off between the passband and the stopband. A lower Q-factor, on the other hand, results in a smoother transition between the passband and the stopband. Chebyshev vs. Butterworth: Speed and Performance The Chebyshev filter is generally faster than the Butterworth filter due to its equiripple frequency response. This equiripple response allows the Chebyshev filter to achieve a steeper roll-off between the passband and the stopband with fewer filter coefficients. Consequently, the filter requires fewer calculations, resulting in faster signal processing. The Butterworth filter, in contrast, is characterized by a maximally flat frequency response in the passband, which results in a slower roll-off between the passband and the stopband. This means that more filter coefficients are required to achieve the desired level of attenuation, leading to slower signal processing. Trade-offs: Quality Factor and Filter Performance The primary trade-off between the Chebyshev and Butterworth filters lies in the balance between the quality factor and the filter's performance. The Chebyshev filter boasts a higher quality factor, which translates to a steeper roll-off and better selectivity. However, this comes at the expense of ripples in the frequency response, which can introduce distortion or signal artifacts. The Butterworth filter, with its maximally flat passband, provides a smoother frequency response with no ripples. This results in lower distortion and signal artifacts but a lower quality factor, which means the filter may struggle to separate closely spaced frequencies. Is the Trade-off Worth It? Deciding whether the trade-off between the quality factor and filter performance is worth it ultimately depends on your specific application and signal processing requirements. If your primary concern is speed and selectivity, the Chebyshev filter may be the better choice. Its higher quality factor and faster signal processing make it an excellent option for applications where steep roll-offs and rapid response times are critical. However, if minimizing signal distortion and artifacts is more important, the Butterworth filter may be more suitable. Its smooth, ripple-free frequency response ensures a cleaner output signal, even if it comes at the cost of a slower roll-off and reduced selectivity. Conclusion: When choosing between the Chebyshev and Butterworth filters, it's essential to consider the balance between speed, quality factor, and filter performance. The Chebyshev filter offers a faster response and a higher quality factor, making it ideal for applications where selectivity and rapid response are crucial. However, its equiripple frequency response can introduce distortion, which may not be suitable for all applications. On the other hand, the Butterworth filter provides a smoother, ripple-free frequency response, but with a lower quality factor and slower roll-off. Ultimately, selecting the right filter for your trading strategy depends on your specific needs and goals. In the world of trading, making timely and accurate decisions is crucial, and the filter you choose plays a significant role in achieving this. Carefully consider the trade-offs between the speed, quality factor, and filter performance when deciding between the Chebyshev and Butterworth filters. By understanding the strengths and weaknesses of each filter type, you can choose the one that best suits your trading requirements and achieve the desired results in your market analysis. Remember that the best filter choice might vary from one trading strategy to another, so always be prepared to reassess your decision based on the unique demands of each trading approach and market conditions.
Education
by The_Peaceful_Lizard
Digital Filters And DSPIntroduction Digital signal processing (dsp) is used to manipulate signals and extract information from them. Among all the tools available, digital filters are the most common ones to use, in fact you are certainly using one during your analysis. In this post i will share some of the knowledge i have associated with filters and their properties. I will try to be the more simple possible with my explanations for those with a light mathematical background :) Before starting talking about filters, lets see some elementary things about dsp, this will help you understand what we will see next. Time Domain It is possible to see signals in different domains. The time domain show the signal with respect to time. For example when you see a price chart you see the price with respect to time, so you are on the time domain. A signal in the time domain is "one dimensional" because there is only one independent variables. Frequency Domain Frequency domain is another way to see a signal, this show the signal with respect to a frequency, in fact you are seeing the frequency content of the signal. In order to understand this i will introduce you to another powerful tool of digital signal processing, the "Fourier Transform". Lets look at some graphical explanations. Lets take a signal we will call S This signal is actually the sum of 3 Sine waves of different Frequencies The one in red have a frequency of 0.07, the one in yellow have a frequency of 0.04 and the one in purple a frequency of 0.02, frequency is defined as the number of cycles in 1 unit time and is expressed in hertz Hz , here we will use "price bar" as unit time, it will be easier to see it this way. The Period is the reciprocal of the frequency, its the number of price bars in 1 cycle. To convert a frequency to a period just divide 1 by the frequency. The amplitude is the maximum absolute value of the signal, often said peak amplitude, for our sine waves the amplitude is 1. So what does the Fourier transform ? Well the Fourier transform convert a signal in the time domain to the frequency domain. In other terms it decompose our signal in a sum of sine waves of different frequency and amplitude.In our signal S its Fourier transform would look like this : Each bar is called a frequency bin or frequency component, they show the frequency and amplitude of one sine wave, each color represent the color of each sine waves i showed earlier. The abscissa x represent the frequency, the ordinate y represent the amplitude. Frequency bins on the left have a lower frequency than those in the right. If we sum each sine waves with the frequency and amplitude each bins give us we would get back our signal S Its important to know this since filters interact with the frequency content, and the Fourier transform can tell us how. What Is A Filter ? A filter is processing tool that allow us to modify the frequency content of a signal, this mean that if we filter a signal, we will remove or change the amplitude of each frequency bin. Interaction With The Frequency Domain We will later see filters in the time domain but lets see them in frequency domain for the moment. In order to see how a filter interact with the frequency content of a signal we must look at the frequency response of the filter. A frequency response look like this : The frequency response is shown in the frequency domain. So what does the red curve represent ? Well this curve show us how the filter interact with each frequency component, lets look at some terms of the frequency response. The pass band area let each frequency components in that area unchanged, the stop band attenuation or transition band area only change the amplitude of each frequency components in that area, the stop band area remove each frequency components in that area. For example : Here we have the frequency domain of a signal, after passing a filter to this signal the frequency domain will look like this : In the image showed earlier you have a blue line, this line delimit the boundary between the pass band and the stop band attenuation, and this boundary is named cutoff frequency , the cutoff frequency is a value expressed in hertz. Actually when you are using a moving average, the length parameter control the position of the cutoff, but we will see moving averages later in more depth. Types Of Filters Now that we understand how a filter interact with frequency domain we must talk about different types of filters, filters can interact in different ways with the frequency components. Low Pass Filters : The name is straightforward, those filters let pass low frequency components and remove the high ones. Frequency response of low pass filters look like this : High Pass Filters : They let pass high frequency components while removing low ones. Bandpass Filters : A mix of the two previous filters, they let pass frequency components in a certain range. Those can have two-cutoff f1, f2 with bandwidth f2 - f1 or one cutoff and one bandwidth parameter. Band-Stop Filters : Sometime called Notch filters or band-reject, they do the contrary of the bandpass filter, instead of letting pass frequency components in a certain range they remove them. Design Of Different Types Of Filters Low Pass Filters : Low pass filters are used to remove high frequencies, when you remove high frequency components of a signal the result is a smoother signal, high frequency create irregularities in a signal and create roughness this is why the result is smoother. Low pass filters are also the core of the design of other filter types. The simplest low pass filter is called a simple moving average or running/rolling - mean/average, or boxcar filter, you will understand why the name "Boxcar" later. The average value or mean of a signal is the sum of all values divided by the number of values, a moving average instead of using all the values will use a proportion of them.A simple moving average can be computed in different ways, here are the easiest ones. 1/n * sum(S,n) sum(1/n * S,n) sum(S,n)/n change(cum(S),n)/n where S is the signal we want to filter and n the period or number of data points we want to average. The frequency response of a simple moving average look like this : The higher the period the more little bumps in the stop band will be created, this is why the moving average still appear a bit rough, its because not all the frequencies in the stop band are removed, some are just attenuated. Those "bumps" are called ripples, when they are situated in the pass band we call them pass band ripples, when they are in the stop band we call them stop band ripples. We will later see filters who does not have ripples. High Pass Filters They remove low frequencies of a signal, its really easy to calculate those, the only thing you have to do is subtract your signal S with a low pass filter, the result show all the signal who have been filtered by the low pass filters. The frequency response of the low pass filter is then reversed. Here the frequency response of a high pass moving average filter : They are really useful because its quite hard to forecast trends in a signal, but its way easier to forecast mid term cycles. Bandpass Filters They let pass frequencies in a range and they are also really easy to compute, first compute a high-pass filter by subtracting the price to a moving average low pass filter, then apply a low pass filter to the result and you end up with a bandpass filter, easy isn't it ? When creating a bandpass filter you can get the pictures of the frequency response of the filter during each step. First step -> high pass filtering Second Step -> Low pass filtering of the first step result Band-Stop Filter They are the opposite of the bandpass filter, so in order to construct those you just need you signal S and your signal S filtered by a bandpass filter. Subtract your signal S with the bandpass filtered version and you get a band-stop filter. Filter Lag Lag is defined as the effect of filters to show past frequencies, this delay is quite unfortunate for us since a filter with zero lag would violate causality and could provide what is called "The Holy Grail". However it is possible to minimize lag, one way is to amplify a frequency range of the signal and then filter the result, a filter describing this process is the zero-lag exponential moving average in the form of : L = (Period - 1)/2 Y0 = S + (S - S(L)) -> Amplification Process Y1 = EMA(Y0,Period) -> Filtering Process Other methods exist but at the end getting optimal smoothing with minimum lag is not something classical filters can do. This is why adaptive filters such as Kalman and Wiener filters where made, but this is another subject that is more complicated. Impulse Response We are entering a really interesting topic, the topic of impulses responses and convolution. First what is an impulse ? An impulse is just defined as a really brief signal, here an example The impulse response denoted h(t) is the reaction of a system to an impulse, so the impulse response of a filter is the filter using an impulse an input. Now lets see the impulse response of a simple moving average of period 5 (n = 5) , you will understand why they are called boxcar filters. its just a "box", puns apart there is also something more than incredible with impulse responses, do you remember how to construct a moving average ? One of the ways was : sum(1/n * S,n) , 1/5 = 0.2, now look at the impulse response from closer Its 0.2, or 1/5, nice isn't it ? When the impulse is equal to 1, the impulse response of a filter h(t) give us values called filter coefficients and this introduce the concept of convolution. Convolution The subject is vast but in resume convolution is the process of combining two function to create a third function. The symbol of convolution is ∗ not to be confounded with * who is used for multiplication. To make it easier to understand convolution lets take back our 5 period moving average case and its impulse response h(t) . Now lets make an operation describing convolution S ∗ h(t) = sum(S * h(t),5) This operation give us a moving average of period 5. So impulse responses can give you really useful information about your filter. IIR Filters There are 2 filter family, F inite I mpulse R esponse filters or FIR and I nfinite I mpulse R esponse filters. IIR filters use recursion, this mean that they use output values as input, if we check the impulse response of those filter h(t) you will see that h(t) goes on forever after the impulse, this is why they are called infinite impulse response. The most well known IIR filter is the exponential moving average (ema), or exponential averager which is described as : y = aS(t) + (1-a)y(t-1) where a is called a smoothing constant and 0 < a < 1 . An IIR filter can also be in the more common form : y(t) = a0 * S(t) + a1 * S(t-1) + b1 * y(t-1) Advanced Filter Design Based on which filter you want to design (FIR/IIR) you will use different methods. FIR filters are easier to design, first select what frequency response your filter should have, then compute the Fourier transform of this frequency response, this will give you the impulse response and therefore the coefficients of the filter you want to design, then you only need to apply convolution. So basically, the Fourier transform of the filter impulse response give you the frequency response of the filter and vice versa. IIR filters are way more complicated to design, you will need the z-transform and this is something that i don't really know about, how unfortunate. Conclusions I hope it wasn't that complicated because it shouldn't be, of course you can make filters without knowing all this, but its always cool to have more information. If you don't understand something or think i have made an error somewhere don't hesitate to mention it in the comment :) Thanks for reading !
Education
by alexgrover
19