June 20, 2023

# What is filter order?

In signal processing, a filter is a device that removes unwanted components or features from a signal. The order of a filter is an essential parameter that determines the extent to which certain signals are reduced or eliminated. This article will explore the concept of filter order and its significance in the field of signal processing.

## Understanding filter order

When designing digital filters, it is essential to specify the filter order. The filter order refers to the number of poles or zeros in the filter transfer function. The order of the filter depends on the number of frequency points at which the filter has a significant effect on the signal. The greater the order of the filter, the more significant the effect on the signal.

### Definition of filter order

The filter order is the number of reactive elements in the circuit that affects the filter transfer function. These reactive elements of the filter refer to capacitors, resistors, and inductors. In practice, the order of the filter is calculated as the polynomial degree of the transfer function.

It is important to note that the filter order is not the same as the filter type. The filter type refers to the mathematical algorithm used to design the filter, such as Butterworth, Chebyshev, or Elliptic filters. The filter order, on the other hand, is a characteristic of the filter that determines its frequency response.

### Importance of filter order in signal processing

The filter order is an essential parameter in signal processing. It defines the ability of the filter to attenuate or suppress specific frequency ranges. Further, the filter order determines the sharpness of the transition band between the passband and stopbands.

For example, a low-order filter with a cutoff frequency of 100 Hz may not effectively remove noise at 150 Hz, whereas a higher-order filter with the same cutoff frequency may attenuate the noise more effectively. Additionally, a higher-order filter can provide a steeper transition band, which means that the filter can more effectively separate the desired signal from unwanted noise or interference.

However, increasing the filter order comes at a cost. Higher-order filters require more complex circuitry and more processing power, which can increase the cost and complexity of the overall system. Furthermore, higher-order filters can introduce phase distortion, which can affect the fidelity of the filtered signal.

Therefore, it is important to carefully consider the filter order when designing a filter for a particular application. The filter order should be chosen to provide the required frequency response while minimizing cost and complexity and avoiding unwanted phase distortion.

## Types of filters and their order

Filters are an essential component of signal processing systems and are used to remove unwanted frequencies or noise from a signal. Various types of filters exist, such as low-pass, high-pass, band-pass, and band-stop filters, which all have different orders. The order of a filter refers to the number of reactive components, such as capacitors and inductors, in the filter circuit.

### Low-pass filters

A low-pass filter attenuates signals above the cutoff frequency and allows frequencies below the cutoff frequency to pass through. The cutoff frequency is the frequency at which the filter starts to attenuate the signal. Low-pass filters are commonly used in audio systems to remove high-frequency noise, such as hissing or static, from the signal. The order of the filter determines the sharpness of the cutoff frequency. A higher-order filter will have a steeper cutoff slope and will provide better attenuation of unwanted frequencies.

Low-pass filters are also used in radio communication systems to prevent interference from adjacent channels. By using a low-pass filter, the frequency range of the transmitted signal can be limited, reducing the potential for interference from other channels.

### High-pass filters

A high-pass filter attenuates frequencies below the cutoff frequency and allows signals above the cutoff frequency to pass through. High-pass filters are commonly used in audio systems to remove low-frequency noise, such as hum or rumble, from the signal. The order of the high-pass filter determines the sharpness of the cutoff frequency. A higher-order filter will have a steeper cutoff slope and will provide better attenuation of unwanted frequencies.

High-pass filters are also used in image processing systems to enhance images by removing low-frequency noise, such as blur or haze, from the image. By using a high-pass filter, the high-frequency components of the image can be enhanced, resulting in a sharper and clearer image.

### Band-pass filters

A band-pass filter is a type of filter that allows signals within a particular frequency range to pass through. The frequency range between the lower and upper cutoff frequencies is known as the passband. The order of the filter determines the bandwidth of the frequency range that the filter passes. A higher-order filter will have a narrower passband and better attenuation of unwanted frequencies outside the passband.

Band-pass filters are commonly used in radio communication systems to select a specific frequency range for transmission or reception. By using a band-pass filter, the system can reject unwanted frequencies outside the passband, reducing interference and improving the signal-to-noise ratio.

### Band-stop filters

A band-stop filter attenuates signals within a particular frequency range. Also known as a notch filter, the band-stop filter allows all other frequencies to pass through. The frequency range between the lower and upper cutoff frequencies is known as the stopband. The order of the filter determines the sharpness or width of the attenuation band. A higher-order filter will have a narrower stopband and better attenuation of unwanted frequencies within the stopband.

Band-stop filters are commonly used in audio systems to remove specific frequencies, such as a 60 Hz hum caused by electrical interference. By using a band-stop filter, the unwanted frequency can be attenuated, resulting in a cleaner and clearer audio signal.

Overall, the choice of filter type and order depends on the specific application and the desired level of attenuation of unwanted frequencies.

## Factors affecting filter order

When it comes to designing a filter, there are a number of factors that come into play. While the filter order is an important consideration, it is not the only one. Let's take a closer look at some of the key factors that can affect the order of a filter.

### Filter specifications

One of the most important factors that can affect the order of a filter is the set of specifications that the filter must meet. These specifications can include things like the cutoff frequency, passband ripple, stopband attenuation, and transition width. In general, the more stringent the specifications, the higher the order of the filter will need to be in order to meet them.

For example, if a filter needs to have a very sharp cutoff at a particular frequency, then it will likely require a higher order filter than if the cutoff can be more gradual. Similarly, if the passband ripple or stopband attenuation need to be very low, then a higher order filter will be required to achieve this.

### Desired frequency response

Another factor that can affect the order of a filter is the desired frequency response. This refers to how the filter should behave in terms of passing or rejecting certain frequencies. For example, if a filter needs to have a very steep transition between the passband and stopband, then a higher order filter will be required to achieve this. On the other hand, if a more gradual transition is acceptable, then a lower order filter may be sufficient.

In addition to the transition width, other aspects of the frequency response can also affect the filter order. For example, if a filter needs to have a very flat passband or stopband, then a higher order filter may be required to achieve this.

### Filter design techniques

The design technique that is used to create a filter can also have a significant impact on its order. Different design techniques, such as Butterworth, Chebyshev, and Elliptic, have different orders and affect the filter design differently.

For example, Butterworth filters are known for having a very flat passband response, but a slower roll off than other filter types. Chebyshev filters, on the other hand, have a steeper roll off but a ripple in the passband. Elliptic filters have a very steep roll off and a flat passband, but have ripples in both the passband and stopband.

Each of these design techniques has its own strengths and weaknesses, and the choice of technique will depend on the specific requirements of the filter design.

As you can see, there are a number of factors that can affect the order of a filter. By considering these factors carefully, it is possible to design a filter that meets the necessary specifications while minimizing its order and complexity.

## Calculating filter order

The determination of filter order is a crucial aspect of filter design. Several methods exist for determining the order of a filter. The filter order is directly proportional to its complexity and the number of components required for its construction. Therefore, it is essential to determine the filter order accurately to minimize the cost and complexity of the filter.

### Methods for determining filter order

The filter order can be determined using graphical methods such as the Bode plot or Nyquist plot. These methods offer a visual representation of the filter's behavior. However, they have limitations in accurately determining the filter order. The graphical methods are suitable for simple filters with a small number of components. For complex filters, mathematical methods are more accurate and efficient.

Alternatively, mathematical methods such as the frequency sampling method and the windowing approach can accurately calculate the filter order. The frequency sampling method involves sampling the desired frequency response of the filter and then using the inverse Fourier transform to obtain the impulse response. The order of the filter is then determined based on the length of the impulse response. The windowing approach involves multiplying the ideal impulse response with a window function to obtain a finite impulse response filter. The order of the filter is determined based on the length of the windowed impulse response.

### Examples of filter order calculations

Consider a low-pass filter with a cutoff frequency of 1kHz and a passband ripple of 0.1dB. Suppose we desire a stopband attenuation of 40dB and a transition width of 150Hz. Using the Butterworth design technique, the filter's order is calculated as 5. The Butterworth filter is a type of low-pass filter that has a maximally flat frequency response in the passband. The Butterworth filter is widely used due to its simplicity and good performance.

Similarly, a Chebyshev type I filter with a transition width of 20Hz, a passband ripple of 1dB, and a stopband attenuation of 40dB has an order of 7. The Chebyshev filter is a type of filter that has a steeper roll-off than the Butterworth filter. The Chebyshev filter has ripple in the passband, which can be controlled by adjusting the design parameters.

In conclusion, determining the filter order is an essential step in filter design. The choice of method for determining the filter order depends on the complexity of the filter and the desired accuracy. Graphical methods are suitable for simple filters, while mathematical methods are more accurate and efficient for complex filters.

## Conclusion

In conclusion, filter order is an essential aspect of digital filter design. The order of the filter determines the filter's behavior regarding signal attenuation and frequency response. Several types of filters exist, each with its order, and different factors such as filter specifications, desired frequency response, and design technique affect the filter order. Various methods exist for determining the filter order, such as graphical and mathematical approaches. Accurate determination of filter order is crucial to achieving the desired frequency response and behavior of the filter.