The transfer function is also called the system function .
It remains to define ``z transform'', and to prove that the z transform of the impulse response always gives the transfer function, which we will do by proving the convolution theorem for z transforms.
The bilateral z transform of the discrete-time signal is defined to be
where is a complex variable. Since signals are typically defined to begin (become nonzero) at time , and since filters are often assumed to be causal,7.1 the lower summation limit given above may be written as 0 rather than to yield the unilateral z transform:
The unilateral z transform is most commonly used. For inverting z transforms, see §6.8.
Recall (§4.1) that the mathematical representation of a discrete-time signal maps each integer to a complex number ( ) or real number ( ). The z transform of , on the other hand, , maps every complex number to a new complex number . On a higher level, the z transform, viewed as a linear operator, maps an entire signal to its z transform . We think of this as a ``function to function'' mapping. We may say is the z transform of by writing
The z transform of a signal can be regarded as a polynomial in , with coefficients given by the signal samples. For example, the signal
The z transform of a finite-amplitude signal will always exist provided (1) the signal starts at a finite time and (2) it is asymptotically exponentially bounded, i.e., there exists a finite integer , and finite real numbers and , such that for all . The bounding exponential may even be growing with (). These are not the most general conditions for existence of the z transform, but they suffice for most practical purposes.
For a signal growing as , for , one would naturally expect the z transform to be defined only in the region of the complex plane. This is expected because the infinite series
More generally, it turns out that, in all cases of practical interest, the domain of can be extended to include the entire complex plane, except at isolated ``singular'' points7.2 at which approaches infinity (such as at when ). The mathematical technique for doing this is called analytic continuation, and it is described in §D.1 as applied to the Laplace transform (the continuous-time counterpart of the z transform). A point to note, however, is that in the extension region (all points such that in the above example), the signal component corresponding to each singularity inside the extension region is ``flipped'' in the time domain. That is, ``causal'' exponentials become ``anticausal'' exponentials, as discussed in §8.7.
Shift and Convolution Theorems
The shift theorem says that a delay of samples in the time domain corresponds to a multiplication by in the frequency domain:
where we used the causality assumption for .
The convolution theorem provides a major cornerstone of linear systems theory. It implies, for example, that any stable causal LTI filter (recursive or nonrecursive) can be implemented by convolving the input signal with the impulse response of the filter, as shown in the next section.
where ``'' means convolution as before. Taking the z transform of both sides of Eq.(6.3) and applying the convolution theorem from the preceding section gives
where H(z) is the z transform of the filter impulse response. We may divide Eq.(6.4) by to obtain
Since z transforming the convolution representation for digital filters was so fruitful, let's apply it now to the general difference equation, Eq.(5.1). To do this requires two properties of the z transform, linearity (easy to show) and the shift theorem (derived in §6.3 above). Using these two properties, we can write down the z transform of any difference equation by inspection, as we now show. In §6.8.2, we'll show how to invert by inspection as well.
Let's take the z transform of both sides, denoting the transform by . Because is a linear operator, it may be distributed through the terms on the right-hand side as follows:7.3 where we used the superposition and scaling properties of linearity given on page , followed by use of the shift theorem, in that order. The terms in may be grouped together on the left-hand side to get
Factoring out the common terms and gives
the z transform of the difference equation yields
Thus, taking the z transform of the general difference equation led to a new formula for the transfer function in terms of the difference equation coefficients. (Now the minus signs for the feedback coefficients in the difference equation Eq.(5.1) are explained.)
The numerator roots are called the zeros of the transfer function, and the denominator roots are called the poles of the filter. Poles and zeros are discussed further in Chapter 8.
Series and Parallel Transfer Functions
- Transfer functions of filters in series multiply together.
- Transfer functions of filters in parallel sum together.
Figure 6.2 illustrates the parallel combination of two filters. The filters and are driven by the same input signal , and their respective outputs and are summed. The transfer function of the parallel combination is therefore
Series Combination is Commutative
Since multiplication of complex numbers is commutative, we have
An important tool for inverting the z transform and converting among digital filter implementation structures is the partial fraction expansion (PFE). The term ``partial fraction expansion'' refers to the expansion of a rational transfer function into a sum of first and/or second-order terms. The case of first-order terms is the simplest and most fundamental:
and . (The case is addressed in the next section.) The denominator coefficients are called the poles of the transfer function, and each numerator is called the residue of pole . Equation (6.7) is general only if the poles are distinct. (Repeated poles are addressed in §6.8.5 below.) Both the poles and their residues may be complex. The poles may be found by factoring the polynomial into first-order terms,7.4e.g., using the roots function in matlab. The residue corresponding to pole may be found analytically as
when the poles are distinct. The matlab function residuez7.5 will find poles and residues computationally, given the difference-equation (transfer-function) coefficients.
Note that in Eq.(6.8), there is always a pole-zero cancellation at . That is, the term is always cancelled by an identical term in the denominator of , which must exist because has a pole at . The residue is simply the coefficient of the one-pole term in the partial fraction expansion of at . The transfer function is , in the limit, as .
Consider the two-pole filter
We thus conclude that
PFE to Real, Second-Order Sections
When all coefficients of and are real (implying that is the transfer function of a real filter), it will always happen that the complex one-pole filters will occur in complex conjugate pairs. Let denote any one-pole section in the PFE of Eq.(6.7). Then if is complex and describes a real filter, we will also find somewhere among the terms in the one-pole expansion. These two terms can be paired to form a real second-order section as follows:
Expressing the pole in polar form as , and the residue as , the last expression above can be rewritten as
Expanding a transfer function into a sum of second-order terms with real coefficients gives us the filter coefficients for a parallel bank of real second-order filter sections. (Of course, each real pole can be implemented in its own real one-pole section in parallel with the other sections.) In view of the foregoing, we may conclude that every real filter with can be implemented as a parallel bank of biquads.7.6 However, the full generality of a biquad section (two poles and two zeros) is not needed because the PFE requires only one zero per second-order term.
To see why we must stipulate in Eq.(6.7), consider the sum of two first-order terms by direct calculation:
Notice that the numerator order, viewed as a polynomial in , is one less than the denominator order. In the same way, it is easily shown by mathematical induction that the sum of one-pole terms can produce a numerator order of at most (while the denominator order is if there are no pole-zero cancellations). Following terminology used for analog filters, we call the case a strictly proper transfer function.7.7 Thus, every strictly proper transfer function (with distinct poles) can be implemented using a parallel bank of two-pole, one-zero filter sections.
Inverting the Z Transform
In the improper case, discussed in the next section, we additionally obtain an FIR part in the z transform to be inverted:
The case of repeated poles is addressed in §6.8.5 below.
FIR Part of a PFE
An alternate FIR part is obtained by performing long division on the reversed polynomial coefficients to obtain
where is again the order of the FIR part. This type of decomposition is computed (as part of the PFE) by residued, described in §J.6 and illustrated numerically in §6.8.8 below.
We may compare these two PFE alternatives as follows: Let denote , , and . (I.e., we use a subscript to indicate polynomial order, and `' is omitted for notational simplicity.) Then for we have two cases:
In the first form, the coefficients are ``left justified'' in the reconstructed numerator, while in the second form they are ``right justified''. The second form is generally more efficient for modeling purposes, since the numerator of the IIR part ( ) can be used to match additional terms in the impulse response after the FIR part has ``died out''.
In summary, an arbitrary digital filter transfer function with distinct poles can always be expressed as a parallel combination of complex one-pole filters, together with a parallel FIR part when . When there is an FIR part, the strictly proper IIR part may be delayed such that its impulse response begins where that of the FIR part leaves off.
In artificial reverberation applications, the FIR part may correspond to the early reflections, while the IIR part provides the late reverb, which is typically dense, smooth, and exponentially decaying . The predelay (``pre-delay'') control in some commercial reverberators is the amount of pure delay at the beginning of the reverberator's impulse response. Thus, neglecting the early reflections, the order of the FIR part can be viewed as the amount of predelay for the IIR part.
The general second-order case with (the so-called biquad section) can be written when as
The delayed form of the partial fraction expansion is obtained by leaving the coefficients in their original order. This corresponds to writing as a ratio of polynomials in :
Numerical examples of partial fraction expansions are given in §6.8.8 below. Another worked example, in which the filter is converted to a set of parallel, second-order sections is given in §3.12. See also §9.2 regarding conversion to second-order sections in general, and §G.9.1 (especially Eq.(G.22)) regarding a state-space approach to partial fraction expansion.
Alternate PFE Methods
Another method for finding the pole residues is to write down the general form of the PFE, obtain a common denominator, expand the numerator terms to obtain a single polynomial, and equate like powers of . This gives a linear system of equations in unknowns , .
Finally, one can alternatively construct a state space realization of a strictly proper transfer function (using, e.g., tf2ss in matlab) and then diagonalize it via a similarity transformation. (See Appendix G for an introduction to state-space models and diagonalizing them via similarity transformations.) The transfer function of the diagonalized state-space model is trivially obtained as a sum of one-pole terms--i.e., the PFE. In other words, diagonalizing a state-space filter realization implicitly performs a partial fraction expansion of the filter's transfer function. When the poles are distinct, the state-space model can be diagonalized; when there are repeated poles, it can be block-diagonalized instead, as discussed further in §G.10.
When poles are repeated, an interesting new phenomenon emerges. To see what's going on, let's consider two identical poles arranged in parallel and in series. In the parallel case, we have
Dealing with Repeated Poles Analytically
and the three residues associated with the pole are 1, 2, and 4.
Let denote the th residue associated with the pole , . Successively differentiating times with respect to and setting isolates the residue :
For the example of Eq.(6.12), we obtain
Impulse Response of Repeated Poles
In the time domain, repeated poles give rise to polynomial amplitude envelopes on the decaying exponentials corresponding to the (stable) poles. For example, in the case of a single pole repeated twice, we have
Proof: First note that
Note that is a first-order polynomial in . Similarly, a pole repeated three times corresponds to an impulse-response component that is an exponential decay multiplied by a quadratic polynomial in , and so on. As long as , the impulse response will eventually decay to zero, because exponential decay always overtakes polynomial growth in the limit as goes to infinity.
In the previous section, we found that repeated poles give rise to polynomial amplitude-envelopes multiplying the exponential decay due to the pole. On the other hand, two different poles can only yield a convolution (or sum) of two different exponential decays, with no polynomial envelope allowed. This is true no matter how closely the poles come together; the polynomial envelope can occur only when the poles merge exactly. This might violate one's intuitive expectation of a continuous change when passing from two closely spaced poles to a repeated pole.
To study this phenomenon further, consider the convolution of two one-pole impulse-responses and :
The finite limits on the summation result from the fact that both and are causal. Recall the closed-form sum of a truncated geometric series:
Going back to Eq.(6.14), we have
which is the first-order polynomial amplitude-envelope case for a repeated pole. We can see that the transition from ``two convolved exponentials'' to ``single exponential with a polynomial amplitude envelope'' is perfectly continuous, as we would expect.
We also see that the polynomial amplitude-envelopes fundamentally arise from iterated convolutions. This corresponds to the repeated poles being arranged in series, rather than in parallel. The simplest case is when the repeated pole is at , in which case its impulse response is a constant:
Alternate Stability Criterion
In §5.6 (page ), a filter was defined to be stable if its impulse response decays to 0 in magnitude as time goes to infinity. In §6.8.5, we saw that the impulse response of every finite-order LTI filter can be expressed as a possible FIR part (which is always stable) plus a linear combination of terms of the form , where is some finite-order polynomial in , and is the th pole of the filter. In this form, it is clear that the impulse response always decays to zero when each pole is strictly inside the unit circle of the plane, i.e., when . Thus, having all poles strictly inside the unit circle is a sufficient criterion for filter stability. If the filter is observable (meaning that there are no pole-zero cancellations in the transfer function from input to output), then this is also a necessary criterion.
A transfer function with no pole-zero cancellations is said to be irreducible. For example, is irreducible, while is reducible, since there is the common factor of in the numerator and denominator. Using this terminology, we may state the following stability criterion:
This characterization of stability is pursued further in §8.4, and yet another stability test (most often used in practice) is given in §8.4.1.
Summary of the Partial Fraction Expansion
In summary, the partial fraction expansion can be used to expand any rational z transform
for , and
for , where the term is optional, but often preferred. For real filters, the complex one-pole terms may be paired up to obtain second-order terms with real coefficients. The PFE procedure occurs in two or three steps:
- When , perform a step of long division to obtain an FIR part and a strictly proper IIR part .
- Find the poles , (roots of ).
- If the poles are distinct, find the residues ,
- If there are repeated poles, find the additional residues via
the method of §6.8.5, and the general form of the PFE is
where denotes the number of distinct poles, and denotes the multiplicity of the th pole.
In step 2, the poles are typically found by factoring the denominator polynomial . This is a dangerous step numerically which may fail when there are many poles, especially when many poles are clustered close together in the plane.
The following matlab code illustrates factoring to obtain the three roots, , :
A = [1 0 0 -1]; % Filter denominator polynomial poles = roots(A) % Filter poles
Software for Partial Fraction Expansion
B = [1 0 0 0.125]; A = [1 0 0 0 0 0.9^5]; [r,p,f] = residuez(B,A) % r = % 0.16571 % 0.22774 - 0.02016i % 0.22774 + 0.02016i % 0.18940 + 0.03262i % 0.18940 - 0.03262i % % p = % -0.90000 % -0.27812 - 0.85595i % -0.27812 + 0.85595i % 0.72812 - 0.52901i % 0.72812 + 0.52901i % % f = (0x0)
For the filter
we obtain the output of residued (§J.6) shown in Fig.6.4. In contrast to residuez, residued delays the IIR part until after the FIR part. In contrast to this result, residuez returns r=[-24;16] and f=[10;2], corresponding to the PFE
in which the FIR and IIR parts have overlapping impulse responses.
B=[2 6 6 2]; A=[1 -2 1]; [r,p,f,m] = residued(B,A) % r = % 8 % 16 % % p = % 1 % 1 % % f = % 2 10 % % m = % 1 % 2
The matlab function conv (convolution) can be used to perform polynomial multiplication. For example:
B1 = [1 1]; % 1st row of Pascal's triangle B2 = [1 2 1]; % 2nd row of Pascal's triangle B3 = conv(B1,B2) % 3rd row % B3 = 1 3 3 1 B4 = conv(B1,B3) % 4th row % B4 = 1 4 6 4 1 % ...The matlab conv(B1,B2) is identical to filter(B1,1,B2), except that conv returns the complete convolution of its two input vectors, while filter truncates the result to the length of the ``input signal'' B2.7.10 Thus, if B2 is zero-padded with length(B1)-1 zeros, it will return the complete convolution:
B1 = [1 2 3]; B2 = [4 5 6 7]; conv(B1,B2) % ans = 4 13 28 34 32 21 filter(B1,1,B2) % ans = 4 13 28 34 filter(B1,1,[B2,zeros(1,length(B1)-1)]) % ans = 4 13 28 34 32 21
Polynomial Division in Matlab
B = [ 2 6 6 2]; % 2*(1+1/z)^3 A = [ 1 -2 1]; % (1-1/z)^2 [firpart,remainder] = deconv(B,A) % firpart = % 2 10 % remainder = % 0 0 24 -8Thus, this example finds that is as written in Eq.(6.21). This result can be checked by obtaining a common denominator in order to recalculate the direct-form numerator:
Bh = remainder + conv(firpart,A) % = 2 6 6 2
firpart = filter(B,A,[1,zeros(1,length(B)-length(A))]) % = 2 10 remainder = B - conv(firpart,A) % = 0 0 24 -8That this must work can be seen by looking at Eq.(6.21) and noting that the impulse-response of the remainder (the strictly proper part) does not begin until time , so that the first two samples of the impulse-response come only from the FIR part.
In summary, we may conveniently use convolution and deconvolution to perform polynomial multiplication and division, respectively, such as when converting transfer functions to various alternate forms.
When carrying out a partial fraction expansion on a transfer function having a numerator order which equals or exceeds the denominator order, a necessary preliminary step is to perform long division to obtain an FIR filter in parallel with a strictly proper transfer function. This section describes how an FIR part of any length can be extracted from an IIR filter, and this can be used for PFEs as well as for more advanced applications .
Frequency Response Analysis
Time Domain Digital Filter Representations