# DSP - FIR Filters, linear phase conditions

• Runei
In summary, the conversation discusses the condition of symmetry or antisymmetry that a FIR filter must follow in order to have a linear phase response. The speaker has been studying this for an exam and has come to understand that a linear phase response is achieved through the formula |H(ω)|X(ω)e^(-jωn_0) which delays the input signal x(n) by n_0 samples. They are now seeking a proof that the impulse response of the filter must be symmetric for a linear phase response to occur."
Runei
Hello there

I'd like to know if anyone has a proof of why the condition of symmetry or antisymmetry must be followed by a FIR filter, in order for it to have a linear phase response?

I've been pouring over this for an exam, and my initial question was what constitutes a linear phase response and after a little pouring and talking with myself I finally got that it is because of the $\left|H\left(\omega\right)\right|$X$\left(\omega\right)e^{-j\omega n_{0}}$ that the x(n) will be delayed by n_0 samples.

Now, however, the only step I need to have rounded this off is a proof to myself that the impulse response of the filter must be symmetric in order for a linear phase response to occur.

Thank you in advance,

Rune

Im just going to bump it once in one last hope :)

## 1. What is a FIR filter?

A Finite Impulse Response (FIR) filter is a type of digital filter used in signal processing. It is characterized by a finite duration of its impulse response, meaning that the output of the filter only depends on a finite number of input samples.

## 2. How does a FIR filter work?

A FIR filter works by convolving the input signal with a finite impulse response sequence of coefficients. These coefficients are chosen to achieve a desired frequency response, such as low-pass, high-pass, or band-pass filtering.

## 3. What are linear phase conditions in FIR filters?

Linear phase conditions refer to the property of a filter where the phase response is a linear function of frequency. This means that all frequency components of the input signal will be delayed by the same amount, preserving the relative timing between them.

## 4. Why are linear phase conditions desirable in FIR filters?

Linear phase conditions are desirable because they preserve the shape of the input signal and do not introduce any phase distortion. This is important in applications where preserving the timing relationships between different frequency components is crucial, such as in audio or image processing.

## 5. How are linear phase conditions achieved in FIR filters?

Linear phase conditions can be achieved in FIR filters by using symmetric impulse response coefficients. This means that the coefficients are mirrored around the midpoint, resulting in an inherently linear phase response. Alternatively, filters with even-order symmetry can also achieve linear phase conditions.

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