# Implementing Impractical Digital Filters

This blog discusses a problematic situation that can arise when
we try to implement certain digital filters. Occasionally in the
literature of DSP we encounter *impractical*
digital IIR filter block diagrams, and by impractical I mean block
diagrams that cannot be implemented. This blog gives examples of
impractical digital IIR filters and what can be done to make them
practical.

**Implementing an Impractical Filter: Example 1**

Reference [1] presented the digital IIR bandpass filter shown in Figure 1(a) where *H*_{NO}(*z*) is a FIR notch filter and variable *K* is a constant.

** Figure 1: Reference [1]'s proposed IIR bandpass filter: (a) high-level depiction; (b) detailed signal path depiction.**

The idea behind this bandpass filter is to subtract the output of the notch filter from the *X*(*z*) input to produce a narrow bandpass filter output. The Figure 1(a) filter's *z*-domain transfer function is:

$$H_{\mathsf{BP}}(z) = \frac{Y(z)}{X(z)} = \frac{1}{1 + KH_{\mathsf{NO}}(z)}\tag{1}$$

where the bandpass filter's*H*

_{NO}(

*z*) term is a 2nd-order FIR notch filter defined by

$$H_{\mathsf{NO}}(z) = 1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}.\tag{2}$$

In Eq. (2) variable $\omega_c$, where $0<{\omega_c}<\pi$, represents the center-frequency of the FIR filter's notch measured in radians/sample.

Based on Eq. (2) the proposed bandpass filter can be depicted as
shown in Figure 1(b). The Figure 1 filter seems like a plausible design
but is in fact impossible to implement (i.e., it's *impractical*). Here's why.

**The Problem**

The trouble with the above bandpass filter is that it contains a feedback signal path containing no delay element (a *delay-less* signal path) as shown by the bold arrows in Figure 2.

** Figure 2: Time-domain depiction of the impractical bandpass filter**** with the delay-less signal path highlighted in bold.**

Upon the arrival of an *x*(*n*) input sample, in order to compute a *y*(*n*) output sample we first need to compute the *w*(*n*) feedback sample. But to compute *w*(*n*) we need *y*(*n*)! That delay-less signal path means the Figure 2 filter cannot be implemented.

**The Solution: Algebra to the Rescue**

As it turns out, algebraic expansion of the *H*_{BP}(*z*) equation produces a new filter *z*-domain transfer function that can be implemented. As shown in Appendix A, we can plug Eq. (2) into Eq. (1) to obtain a *practical* transfer function of:

$$H_{\mathsf{BP,Practical}}(z) = \frac{\alpha}{1+2K{\alpha}\mathsf{cos}(\omega_c)z^{-1}+K{\alpha}z^{-2}}\tag{3}$$

where ${\alpha}=1/(1+K)$, and $K≠-1$. Unlike the Figure 2 filter, the Eq. (3) bandpass filter can be implemented with no problem as shown in Figure 3.

**Figure 3: Practical (implementable) version of the**** impractical Figure 1 bandpass filter.**

**Implementing an Impractical Filter: Example 2**

A second example of an impractical filter, proposed in Reference [2], is the narrowband digital notch filter shown in Figure 4.

**Figure 4: High-level depiction of Reference [2]'s**** proposed IIR notch filter.**

where *C*(*z*) is a notch filter and variable a is a constant. The *z*-domain transfer function for the proposed Figure 4 IIR filter, derived in Appendix B, is:

$$H_\mathsf{N}(z) = \frac{Y(z)}{X(z)} = \frac{(1+\alpha)C(z)}{1+{\alpha}C(z)}.\tag{4}$$

The*C*(

*z*) function is a traditional 2nd-order narrowband IIR notch filter whose transfer function is:

$$C(z) = \frac{1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}}{1-2\rho\mathsf{cos}(\omega_c)z^{-1}+\rho^2z^{-2}}\tag{5}$$

where variable $\omega_c$, $0<{\omega_c}<\pi$, is the center-frequency of the filter's notch measured in radians/sample, and $\rho$ is the radii of the conjugate poles' z-plane locations. Based on Eq. (5) the proposed notch filter can be depicted in more detail as shown in Figure 5.

**Figure 5: Detailed signal path depiction of the*** H*_{N}**( z) notch filter.**

** **

The Figure 5 bandpass filter contains a *delay-less* feedback signal path as shown by the bold arrows in Figure 6.

** Figure 6: Time-domain depiction of the impractical H**

_{N}**(**

*z*)**notch filter with the delay-less signal path**

**highlighted in bold.**

That delay-less signal path makes the Figure 6 filter *impractical* because to compute a *y*(*n*) output sample we first need to compute the *u*(*n*) feedback sample. But to compute *u*(*n*) we need *y*(*n*). Similar to the Figure 2 filter, the Figure 6 filter cannot be implemented.

Again, as we did in Example 1, algebraic expansion of the *H*_{N}(*z*) equation produces a new filter *z*-domain
transfer function that can be implemented. Substituting Eq. (5)'s C(z)
expression into Eq. (4), as shown in Appendix C, we produce a *practical *transfer function of:

$$H_{\mathsf{N,Practical}}(z) = \frac{1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}}{1-2\frac{\rho+\alpha}{1+\alpha}\mathsf{cos}(\omega_c)z^{-1}+\frac{\rho^2+\alpha}{1+\alpha}z^{-2}}\tag{6}$$

where ${\alpha} ≠ -1$. This new *H*_{N,Practical}(*z*) function is a traditional 2nd-order IIR filter that *can* be implemented using the block diagram shown in Figure 7.

**Figure 7: Practical (implementable) version of the**** impractical Figure 4 notch filter.**

**Conclusion**

The bottom line of this blog is: When you encounter a digital network's block diagram that is *impractical*
(i.e., impossible to implement because it contains a delay-less
feedback signal path), algebraic analysis of the network's z-domain
transfer function may well lead to a new and equivalent block diagram
that *is* possible to implement.

**Postscript: December, 2017**

In my original July 2016 version of this blog I presented the following Figure 8 network and claimed it to be impractical.

**Figure 8: Reference [3]'s**** spectrum analysis network.**

But, as was pointed out to me by Prof. Zsolt Kollár (Budapest University of Technology and Economics, Hungary) I was mistaken. If we draw the details of the resonators in Figure 8 we have Figure 9, and that network can indeed be implemented due to the delay elements in each path.

**Figure 9: Expanded version of the Figure 8**** network.**

Upon arrival of an *x*[*n*] input sample we compute all the outputs of the *z** _{k}* multipliers, shift those products through their delay elements, and wait for the next

*x*[

*n*+1] input sample. No problem. As the cowboys in Texas would say, "The Figure 8 system works real fine."

**References**

[1] S.Engelberg, “Precise Variable-Q Filter Design,” *IEEE Signal** Processing Magazine*, Sept. 2008, pp. 113–119.

[2] S. Pei, B, Guo, and W. Lu, "Narrowband Notch Filter Using Feedback Structure", *IEEE Signal Processing Mag**azine*, May 2016, pp. 115-118.

[3] G. Peceli, G. Simon, "Generalization of the Frequency Sampling Method", *IEEE Instr. and Meas. Conference*, Brussels Belgium, June 1996.

https://www.researchgate.net/publication/3632671_Generalization_of_the_frequency_sampling_method

__Appendix A: Derivation of Eq. (3) __

We derive Eq. (3) by starting with the above Eq. (1), repeated here as:

$$H_{\mathsf{BP}}(z) = \frac{1}{1 + KH_{\mathsf{NO}}(z)}.\tag{A-1}$$

Substituting the notch filter's *H*_{NO}(*z*) from Eq. (2) into Eq. (A-1) yields:

$$H_{\mathsf{BP}}(z) = \frac{1}{1 + K[1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}]}$$

$$= \frac{1}{(1 + K)-2K\mathsf{cos}(\omega_c)z^{-1}+Kz^{-2}}.\tag{A-2}$$

Next, dividing Eq. (A-2)'s numerator and denominator by (1+*K*) we produce the desired expression for the Figure 1 proposed bandpass filter's equivalent and *practical* transfer function of:

$$H_{\mathsf{BP,Practical}}(z) = \frac{\alpha}{1+2K{\alpha}\mathsf{cos}(\omega_c)z^{-1}+K{\alpha}z^{-2}}\tag{A-3}$$

where ${\alpha}$ = 1/(1+*K*), and *K* ≠ -1.

__Appendix B: Derivation of Eq. (4) __

The following derivation was graciously supplied to me by Reference [2]'s co-author Bo-Yi Guo. Master's student Guo derived Eq. (4) by redrawing Figure 4 as shown in Figure B1.

**Figure B1: Reference [2]'s proposed IIR notch filter.**

From Figure B1, we start with

$$W(z) = X(z)-Y(z)\tag{B-1}$$

and

$$V(z) = X(z)+{\alpha}W(z).\tag{B-2}$$

Replacing *W*(*z*) in Eq. (B-2) with Eq. (B-1), we have:

$$V(z) = X(z) +{\alpha}(X(z)–Y(z)).\tag{B-3}$$

The *Y*(*z*) output can be expressed as:

$$Y(z) = V(z)C(z).\tag{B-4}$$

Replacing *V*(*z*) in Eq. (B-4) with Eq. (B-3) gives:

$$Y(z) = [X(z) + {\alpha}(X(z)–Y(z))]C(z)$$

*$$= (1+{\alpha})X(z)C(z)-{\alpha}Y(z)C(z).\tag{B-5}$$
*

Rearranging Eq. (B-5) yields our desired *H*_{N}(*z*) transfer function as:

$$H_\mathsf{N}(z) = \frac{Y(z)}{X(z)} = \frac{(1+\alpha)C(z)}{1+{\alpha}C(z)}.\tag{B-6}$$

**Appendix C: Derivation of Eq. (6) **

We derive Eq. (6) by starting with the above Eq. (4) repeated here as:

$$H_\mathsf{N}(z) = \frac{(1+\alpha)C(z)}{1+{\alpha}C(z)}\tag{C-1}$$

or after dividing through by *C*(*z*),

$$H_\mathsf{N}(z) = \frac{1+\alpha}{\frac{1}{C(z)}+\alpha},\tag{C-2}$$

recalling from Eq. (5) that

$$H_{\mathsf{N,Practical}}(z) = \frac{1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}}{1-2\rho\mathsf{cos}(\omega_c)z^{-1}+\rho^2z^{-2}}.\tag{C-3}$$

Next we rewrite Eq. (C-2) by separating *C*(*z*) into its numerator and denominator parts as:

$$H_\mathsf{N}(z) = \frac{1+\alpha}{\frac{1}{C_\mathsf{numer}(z)/C_\mathsf{denom}(z)}+\alpha}= \frac{1+\alpha}{\frac{C_\mathsf{denom}(z)}{C_\mathsf{numer}(z)}+\alpha}$$

$$= \frac{(1+\alpha)C_\mathsf{numer}(z)}{C_\mathsf{denom}(z)+{\alpha}C_\mathsf{numer}(z)}.\tag{C-4}$$

Substituting Eq. (C-3)'s numerator and denominator terms into Eq. (C-4) we have:

$$H_{\mathsf{N}}(z) = \frac{(1+\alpha)[1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}]}{1-2\rho\mathsf{cos}(\omega_c)z^{-1}+\rho^2z^{-2}+\alpha[1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}]}$$

$$= \frac{(1+\alpha)[1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}]}{(1+\alpha)-2(\rho+\alpha)\mathsf{cos}(\omega_c)z^{-1}+(\rho^2+\alpha)z^{-2}}\tag{C-5}$$

which is Reference [2]'s Eq. (15). Finally, dividing Eq. (C-5)'s
numerator and denominator by (1+$\alpha$) we produce the desired
equivalent and *practical* expression for the proposed notch filter's transfer function of:

$$H_{\mathsf{N,Practical}}(z) = \frac{1-2\mathsf{cos}(\omega_c)z^{-1}+z^{-2}}{1-2\frac{\rho+\alpha}{1+\alpha}\mathsf{cos}(\omega_c)z^{-1}+\frac{\rho^2+\alpha}{1+\alpha}z^{-2}}\tag{C-6}$$

where ${\alpha} ≠ -1$.

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