Why fir filter is always stable

FIR Filters:. Impulses to be considered for filtering are finite. There are no feed back Connections from the Output to the Input. Symmetric FIR Filters:. Symmetric FIR Filters have their Impulses that occur as the mirror image in the first quadrant and second quadrant or Third quadrant and fourth quadrant or both. The Antisymmetric FIR Filters have their impulses that occur as the mirror image in the first quadrant and third quadrant or second quadrant and Fourth quadrant or both.

Linear Phase:. The FIR Filters are said to have linear in phase if the filter have the impulses that increases according to the Time in digital domain. Frequency Response:. The Frequency response of the Filter is the relationship between the angular frequency and the Gain of the Filter. Gibbs Phenomenon:.

The abrupt truncation of Fourier series results in oscillation in both passband and stop band. These oscillations are due to the slow convergence of the fourier series. This is termed as Gibbs Phenomenon. Windowing Technique:. To avoid the oscillations instead of truncating the fourier co-efficients we are multiplying the fourier series with a finite weighing sequence called a window which has non-zero values at the required interval and zero values for other Elements.

Total number of bits in x is reduced by using two methods namely Truncation and Rounding. These are known as quantization Processes.

Input Quantization Error:. The Quantized signal are stored in a b bit register but for nearest values the same digital equivalent may be represented. This is termed as Input Quantization Error. Product Quantization Error:. The Multiplication of a b bit number with another b bit number results in a 2b bit number but it should be stored in a b bit register.

This is termed as Product Quantization Error. Co-efficient Quantization Error:. Limit Cycle Oscillations:. If the input is made zero, the output should be made zero but there is an error occur due to the quantization effect that the system oscillates at a certain band of values.

Overflow limit Cycle oscillations:. Overflow error occurs in addition due to the fact that the sum of two numbers may result in overflow. To avoid overflow error saturation arithmetic is used. Dead band:. The range of frequencies between which the system oscillates is termed as Deadband of the Filter.

It may have a fixed positive value or it may oscillate between a positive and negative value. Signal scaling:. The inputs of the summer is to be scaled first before execution of the addition operation to find for any possibility of overflow to be occurred after addition.

The scaling factor s0 is multiplied with the inputs to avoid overflow. Developed by Therithal info, Chennai. Toggle navigation BrainKart. Posted On : Advantages: a FIR filters have exact linear phase. Disadvantages: a For the same filter specifications the order of FIR filter design can be as high as 5 to n10 times that of IIR design. Merits: 1. Demerits: 1. FIR FIlters 6. An interesting situation arises if any poles lie on the unit circle, since the system is said to be marginally stable , as it is neither stable or unstable.

Although marginally stable systems are not BIBO stable, they have been exploited by digital oscillator designers, since their impulse response provides a simple method of generating sine waves, which have proved to be invaluable in the field of telecommunications. The IIR filter implementation discussed herein is said to be biquad , since it has two poles and two zeros as illustrated below in Figure 2. The biquad implementation is particularly useful for fixed point implementations, as the effects of quantization and numerical stability are minimised.

However, the overall success of any biquad implementation is dependent upon the available number precision, which must be sufficient enough in order to ensure that the quantised poles are always inside the unit circle.

Thus, the filtering operation of Figure 1 can be summarised by the following simple recursive equation:. Analysing the equation, notice that the biquad implementation only requires four additions requiring only one accumulator and five multiplications, which can be easily accommodated on any Cortex-M microcontroller.

A collection of Biquad filters is referred to as a Biquad Cascade , as illustrated below. When implementing a filter in floating point i. The Direct Form II Transposed structure is considered the most numerically accurate for floating point implementation, as the undesirable effects of numerical swamping are minimised as seen by analysing the difference equations. The filter summary shown in Figure 4 provides the designer with a detailed overview of the designed filter, including a detailed summary of the technical specifications and the filter coefficients, which presents a quick and simple route to documenting your design.

Although several practical implementations for FIRs exist, the direct form structure and its transposed cousin are perhaps the most commonly used, and as such, all designed filter coefficients are intended for implementation in a Direct form structure.

The Direct form structure and associated difference equation are shown below. The Direct Form is advocated for fixed point implementation by virtue of the single accumulator concept.

The recommended default structure within the ASN Filter Designer is the Direct Form Transposed structure, as this offers superior numerical accuracy when using floating point arithmetic. This can be readily seen by analysing the difference equations below used for implementation , as the undesirable effects of numerical swamping are minimised, since floating point addition is performed on numbers of similar magnitude.

Download demo now. Licencing information. Ask Question. Asked 7 years, 10 months ago. Active 2 years, 10 months ago. Viewed 32k times. Since they contain poles, shouldn't they be more affected by stability issues than others?

Improve this question. Gilles 3, 3 3 gold badges 18 18 silver badges 28 28 bronze badges. Add a comment. Active Oldest Votes. EDIT: After some further thought and some scribbling and google-ing, I think that I have an answer to this question of FIR poles that hopefully will be satisfactory to interested parties. Because they are never beyond the unit circle, they are no threat to the stability of an FIR system. Thus, it is possible to construct FIR filters which have no poles but these filters are then acausal -- i.

Improve this answer. Kenneide Kenneide 1 1 gold badge 8 8 silver badges 10 10 bronze badges. It starts with a simple moving average filter coefficients like 1, 1, 1, 1 and rewrites it recursively - thereby introducing a pole. See link. These are often implemented on FPGAs as the first step in down conversion because in their recursive form they are quite cheap to implement computationally. See the Graychip documentation as an example.

They are usually implemented in fixed point to maintain stability. Show 3 more comments. Thanks for the explanation.