The discrete transfer function of a FIR filter

After seeing its behavior over time, we derive the transfer function of an FIR filter and the conditions for it to introduce a linear phase shift with frequency.

The discrete transfer function of a FIR filter

The transfer function of a block is defined as the transform of its impulse response. This is due to the fact that, in the frequency field, the output signal can be calculated by multiplying the input signal transform by the block transfer function.

Starting from the discrete Fourier transform and applying simplification z=e^{-j2\pi \vartheta } we can define:

ZT[y(n)]=Y(z)=\sum_{n=0}^{\infty }y(n)z^{-n}

This new form in which the discrete Fourier transform can be placed is called the zeta transform.

Without getting, for now, into the topic, (see this previous article), keep in mind that the most important effect of this transformation is that every elementary delay in time, equals in the domain of the transform to a division by z. For example:

ZT[x(n-1)]=X(z)z^{-1}
ZT[x(n-2)]=X(z)z^{-2}

and more generally:

ZT[x(n-p)]=X(z)z^{-p}

Application of the zeta transform to the FIR filter over time

Applying the zeta transform thus defined to the finite difference equation that defines a generic FIR filter over time, (see this previous artile), we get an algebraic equation that constitutes its definition in the complex domain of the variable z.

ZT[y(n)]=ZT[a_{0}x(n)+a_{1}x(n-1)+\cdots +a_{n-1}x(n-N+1)]

or:

Y(z)=a_{0}X(z)+a_{1}X(z)z^{-1}+\cdots +a_{n-1}X(z)z^{-N+1}

At this point it is immediate to derive the filter transfer function:

G(z)=\frac{Y(z)}{X(z)}= a_{0}X(z)+a_{1}X(z)z^{-1}+\cdots +a_{n-1}X(z)z^{-N+1}

that is, in a more compact form:

[1] G(z)=\frac{Y(z)}{X(z)}=\sum_{i=0}^{N-1}a_{i}z^{-i}

The last two equations represent two equivalent ways of writing the discrete transfer function of a generic FIR filter of order N-1.

FIR filters are very important because you can always make sure that they have a linear phase shift with the frequency.

Linear phase shift filters

Although the phase of the transmitted signals is often neglected in monoaural telecommunications systems, such as the telephone, in the case of multiphonic systems it is of fundamental importance in the perception of the position of the sound.

Hence the need for certain applications to have linear phase shift filters with frequency so that the phase is not distorted. This is analogous to what happens in power lines when the Heaviside condition is valid since the physical means have the task of carrying the signal without distorting it.

In fact, if an x(t) signal is transmitted along a line, it is considered not distorting if the Heaviside conditions are respected, that is, if it is admitted that the signal itself can be recovered from the one received y(t) at less than a scale factor and a finite time delay, namely:

y(t)=A_{0}x(t-t_{0})

Considering the revealing property of the pulse, it is easy to verify that this is achieved by seeing the line as a filter whose response to the pulse is worth:

g(t)=A_{0}\delta (t-t_{0})

Therefore, to derive the Heaviside conditions valid in frequency, it is enough to transform the g(t) obtaining:

G(f)=A_{0}e^{-j2\pi ft_{0}}

From this expression we can say that for a filter the Heaviside condition corresponds to the fact that its transfer function has constant amplitude and phase proportional to the frequency for all frequencies; this is completely equivalent to what applies to transmission lines.

Conditions for having a linear phase shift FIR filter

For an FIR filter to introduce a linear phase shift, its coefficients must be symmetrical if the filter is low-pass and antisymmetrical if the filter is high-pass.

Without going into the demonstration, the filter transfer function [1] can be rewritten as:

[2] G(\theta )=e^{-j\pi N\theta }\sum_{k=0}^{\frac{N}{2}-1}[a_{k}cos(2\pi k\theta )+ja_{k}sen(2\pi k\theta )]

where, starting from [1] we placed \theta=fT (normalized frequency), the change of variable k=i-(\frac{N}{2}) and then applied Euler’s formulas.

Now, from [2], we see that if the coefficients ak are symmetric, i.e. ak=a-k, then in the summation the sinusoidal functions disappear and G(θ) will remain real and multiplied by a phase shift factor equal to -\pi N\theta which varies linearly with frequency.

If, on the other hand, the ak coefficients are antisymmetric, i.e. ak=-a-k, the cosinusoidal functions will disappear from the summation. The phase shift will still be linear, due to the presence of the term -\pi N\theta although increased by one factor \frac{\pi }{2} since the function is now purely imaginary. Also note that, in the latter case, G(\theta ) it will become an odd function of the frequency and then the filter will be of the high-pass type.

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