Frequency response of a digital system

Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

Series	Geophysical References Series
Title	Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author	Enders A. Robinson and Sven Treitel
Chapter	15
DOI	http://dx.doi.org/10.1190/1.9781560801610
ISBN	9781560801481
Store	SEG Online Store

We recall that impulse response $h_{k}$ is the system’s response to a unit spike. Thus, the impulse response represents a transient that dies out over time for a stable system. On the other hand, the frequency response can be interpreted as a steady-state response of the system to signals, each with a pure frequency. Let the real variable $\omega$ denote frequency (in radians per second), and let $\Delta t$ be the discrete time spacing (in seconds). The function $e^{i\omega \Delta tn}$ is a steady-state sinusoidal wave of pure frequency $\omega$ If we let this steady-state signal be the input to a unit-delay system Z, the steady-state output is the delayed signal $e^{i\omega \Delta t\left(n-1\right)}$. The frequency response is the ratio of the steady-state output to the steady-state input $e^{-i\omega \Delta t}$; that is,

${\begin{aligned}{\frac {e^{i\omega \Delta t\left(n-1\right)}}{e^{i\omega \Delta tn}}}=e^{-i\omega \Delta t}.\end{aligned}}$

(67)

Thus, the pure-delay operator Z can be represented as the point $Z=e^{-i\omega \Delta t}$ on the unit circle. More generally, if we let the steady-state input $u_{n}=e^{i\omega \Delta tn}$ be the input to the system

${\begin{aligned}y_{n}=\sum _{k=-\infty }^{\infty }{h_{k}}u_{n-k},\end{aligned}}$

(68)

the steady-state output is

${\begin{aligned}y_{n}=\sum _{k=0}^{\infty }{h_{k}}e^{i\omega \Delta t\left(n-k\right)}{=e^{i\omega \Delta tn}\sum _{k=0}^{\infty }{h_{k}}e^{-i\omega \Delta tk}}.\end{aligned}}$

(69)

The frequency response $H\left(e^{-{\rm {i}}\omega \Delta t}\right)$ is defined as the ratio of the steady-state output to the steady-state input; that is, the frequency response is

${\begin{aligned}H(e^{-i\omega \Delta t})={\frac {\sum \limits _{k=0}^{\infty }{h_{k}e^{i\omega \Delta t(n-k)}}}{e^{i\omega \Delta tn}}}=\sum \limits _{k=0}^{\infty }{h_{k}e^{-i\omega \Delta tk}.}\end{aligned}}$

(70)

The frequency response written thus conforms to the standard electrical engineering convention, with negative signs in the exponents. Therefore, although we use the geophysics Z-transform (i.e., the generating function) instead of the electrical engineering z-transform, we use the same convention regarding frequency that electrical engineers use. The transfer function

${\begin{aligned}H\left(Z\right)=\sum _{k=0}^{\infty }{h_{k}}Z^{k}\end{aligned}}$

(71)

of a stable digital system exists in a region that includes the unit circle. If we substitute $Z=e^{-i\omega \Delta t}$ into the transfer function $H\left(Z\right)$, we obtain the frequency response $H\left(e^{-i\omega \Delta t}\right)$. That is, the values of H(Z) on the unit circle make up the frequency response of the system. Instead of the cumbersome notation $H\left(e^{-i\omega \Delta t}\right)$, the frequency response is written more conveniently as $H\left(\omega \right)$. Thus, we have

${\begin{aligned}H(e^{-i\omega \Delta t})=H(\omega )=\sum \limits _{k=-\infty }^{\infty }{h_{k}e^{-i\omega \Delta tk}=A(\omega )e^{i\theta (\omega )}.}\end{aligned}}$

(72)

The frequency response is in the form of a discrete Fourier transform. The magnitude $A\left(\omega \right)$ of the frequency response is called the magnitude spectrum, and the angle $\theta \left(\omega \right)$ is called the phase spectrum — both of which we first encountered in Chapter 6. The phase-lag spectrum $\phi \left(\omega \right)$ is defined as the negative of the phase spectrum; that is, $\varphi \left(\omega \right)=-\theta \left(\omega \right)$. The slope of the phase-lag spectrum is called the group delay; that is, the group delay is $d\phi (\omega )/d\omega$. Because

${\begin{aligned}e^{-i\omega \Delta t}=e^{i\left({\omega +{\frac {2\pi }{\Delta t}}}\right)\Delta t},\end{aligned}}$

(73)

it follows that the frequency response is periodic in $\omega$, with a period of $2\pi /\Delta t$. As a result, we have to examine only the frequency response over one period, which usually is taken from the point $\omega =-\pi /\Delta t$ to the point $\omega =\pi /\Delta t$. The frequency $\omega =\pi /\Delta t$ is known as the Nyquist frequency, which we discussed in Chapter 4. For a causal system, the frequency response is the one-sided discrete Fourier transform

${\begin{aligned}H\left(e^{-i\omega \Delta t}\right)=\sum _{k=0}^{\infty }{h_{k}}e^{-i\omega \Delta tk}.\end{aligned}}$

(74)

The frequency response of an ARMA$\left(p,\ q\right)$ system is

${\begin{aligned}H\left(e^{-i\omega \Delta t}\right)={\frac {{\beta }_{0}+{\beta }_{1}e^{-i\omega \Delta {\rm {t}}}+{\beta }_{2}e^{-i\omega \Delta t2}+\ldots +{\beta }_{q}e^{-i2\omega \Delta tq}}{1+{\alpha }_{1}e^{-i\omega \Delta t}+{\alpha }_{2}e^{-{\rm {i\omega }}\Delta t2}{\rm {+\ldots +}}{\alpha }_{p}e^{-i\omega \Delta tp}}},\end{aligned}}$

(75)

which in factored form is

${\begin{aligned}H\left(e^{-i\omega \Delta t}\right)={\frac {{\beta }_{0}\left(1-b_{1}e^{-i\omega \Delta t}\right)\left(1-b_{2}e^{-i\omega \Delta t}\right)\ldots \left(1-b_{q}e^{-j\omega \Delta t}\right)}{\left(1-a_{1}e^{-i\omega \Delta t}\right)\left(1-a_{2}e^{-i\omega \Delta t}\right)\ldots \left(1-a_{p}e^{-i\omega \Delta t}\right)}}.\end{aligned}}$

(76)

An all-pass system $P\left(z\right)$ is defined as a stable causal system with unit magnitude spectrum; that is,

${\begin{aligned}|P\left(e^{-i\omega \Delta t}\right){\rm {|=1}}\;\;\;\;{\rm {for\ all\ \omega .}}\end{aligned}}$

(77)

All-pass systems can be broken down into four types: trivial, pure-delay, dispersive, and impure-delay systems. The transfer function of a trivial all-pass system is merely a constant of magnitude one. In the case of real systems, this constant must be either +1 or -1, so a (real) trivial all-pass system either produces no phase shift or a 180° phase shift for all frequencies. A pure-delay all-pass system delays a signal by a given positive constant N; that is, it transforms the input $u_{k}$ into $u_{k-{\rm {N}}}$. As a result, the transfer function of a pure-delay all-pass system is

${\begin{aligned}P\left(Z\right)=Z^{N}\mathrm {\;\;\;where\;\;} n>0,\end{aligned}}$

(78)

so the frequency response is

${\begin{aligned}P\left(e^{-i\omega \Delta t}\right)=e^{-i\omega \Delta tN}.\end{aligned}}$

(79)

Thus, a pure-delay all-pass system produces a phase lag of

${\begin{aligned}\psi \left(\omega \right)=\Delta tN\omega ,\end{aligned}}$

(80)

which is a linear function of $\omega$. A dispersive all-pass system is an ARMA$\left(q,\ q\right)$ system of the form

${\begin{aligned}H\left(Z\right)={\frac {\left(-a_{1}^{*}+Z\right)\left(-a_{2}^{*}+Z\right)\ldots \left(-a_{q}^{*}+Z\right)}{\left(1-a_{1}Z\right)\left(1-a_{2}Z\right)\ldots \left(1-a_{q}Z\right)}}.\end{aligned}}$

(81)

In other words, an all-pass system has a special arrangement of poles and zeros: The poles $a_{i}^{-1}$ and the zeros $a_{i^{*}}$ are symmetric with respect to the unit circle. If we write $a_{i}=r_{i}\;\exp \;(-i\vartheta _{i})$, then the pole is $a_{i}^{-1}=r_{i}^{-1}\;\exp \;(i\vartheta _{i})$ and the zero is $a_{i}^{*}=r_{i}\;\exp \;(i\vartheta _{i})$. Both are on the same radial line, with the pole outside the unit circle and the zero inside the unit circle.

Therefore, the frequency spectrum is the product of q factors of the form

${\begin{aligned}{\frac {\left(-a_{i}^{*}+e^{-i\omega \Delta t}\right)}{(1-a_{i}e^{-i\omega \Delta t})}}.\end{aligned}}$

(82)

The frequency spectrum 82 has a magnitude of unity because the complex conjugate of the numerator is equal to ${\rm {\ exp\ }}\left(i\omega \Delta t\right)$ times the denominator. Hence, the numerator has the same magnitude as the denominator.

An impure-delay all-pass system never is used in engineering applications, so we will omit its consideration here.

Let us say a word about notation. We use $P\left(Z\right)$ and $p_{k}$ as the transfer function and the impulse response, respectively, of an all-pass system. This use of p should be kept distinct from the constant p appearing in the designation of an ARMA$\left(p,\ q\right)$ system.

The group-delay theorem (Bode, 1945; Robinson, 1963, theorem 5, p. 204) deals with group delay of a digital all-pass system. The group delay of a nontrivial digital all-pass system is positive for all frequencies.

We can prove the group-delay theorem as follows: Often it is convenient to change our time scale from a scale in seconds to a scale in which the discrete time-spacing $\Delta t$ is one unit. Then we can let $\Delta t=1$ so $Z=e^{-i\omega }$. The Nyquist frequency becomes simply $\pi$, and the Nyquist range becomes $-\pi \leq \omega \leq \pi$. The pure-delay all-pass system, as we have seen, has phase lag $\phi \left(\omega \right)=N\omega$ where time delay $N>0$, so it satisfies the theorem. The dispersive all-pass system is made up of q factors of the form $\left(-a^{*}+Z\right)/\left(1-aZ\right)$. In terms of frequency, this factor becomes

${\begin{aligned}{\frac {-a^{*}+e^{-i\omega }}{1-ae^{-l\omega }}}={\frac {-e^{i\omega }a^{*}{\rm {+l}}}{e^{i\omega }-a}}=-a^{*}{\frac {e^{i\omega }-a^{*-1}}{e^{i\omega }-a}}.\end{aligned}}$

(83)

The constant $-a^{*}$ increases the phase lag by an additive constant, so we can neglect it as far as this proof is concerned. The phase lag $\phi$ of the factor is the difference of the angles of the numerator and denominator; that is,

${\begin{aligned}\phi {\rm {=angle}}\left(e^{i\omega }-(a^{*}\right)^{-1})-{\rm {angle}}\left(e^{i\omega }-a\right).\end{aligned}}$

(84)

In Figure 10, we show the angle $\phi$ for various points on the unit circle. We trace out one period of the frequency function if we travel counterclockwise around the unit circle from point A to B, C, D, E, and back to A. Over this journey, we see that the angle $\phi$ increases monotonically from 0 to $2\pi$. Thus, the given factor of a dispersive all-pass system produces an increase in phase lag of $2\pi$ over one complete period $2\pi$ of frequency $\omega$ Therefore, a dispersive all-pass system of q such factors would have a phase-lag spectrum $\phi \left(\omega \right)$ that increases monotonically by an amount $2\pi q$ over one complete period; that is,

Figure 10.  As the unit circle is traversed counterclockwise from A back to A again, the phase-lag increases from 0 to $2\pi$.

${\begin{aligned}\phi \left(\phi \right)-\psi \left(-\phi \right){\rm {2}}\pi q.\end{aligned}}$

(85)

Because the phase-lag spectrum $\phi \left(\omega \right)$ of an all-pass system is increasing monotonically, it follows that its derivative $d\phi \left(\omega \right)ld\omega$ is positive. This derivative is the group delay. Thus, a nontrivial all-pass system has positive group delay for all frequencies $\omega$.

By definition, an all-pass system is a causal system with the property that the magnitude spectrum of the output signal is the same as the magnitude spectrum of the input signal. The energy-delay theorem (Robinson, 1963, theorem 7, p. 204) establishes that an all-pass system has the property that the time structure of output energy is delayed with respect to the time structure of input energy. In other words, an all-pass system differentially delays the return of energy in passing a signal from input to output. An all-pass system is comparable to a bureaucrat who day by day receives written requests on his desk and who holds them and later passes them on in some other order to a subordinate. Nothing in content has changed, but certain requests are delayed with respect to others. In seismic processing, an all-pass system always gives trouble.

The energy-delay theorem has the formal statement: Let a stable causal signal $u_{k}$ be the input to a nontrivial digital all-pass system $P\left(Z\right)$, and let the stable causal signal $y_{k}$ be the resulting output. The front-end energy of the input is greater than or equal to the front-end energy of the output for all time indices n; that is,

${\begin{aligned}\sum _{k=0}^{n}{u_{k}^{2}}\geq \sum _{k=0}^{n}{y_{k}^{2}}.\end{aligned}}$

(86)

The strict inequality (i.e., greater than) holds for some $n\geq 0$. For $n=\infty$, the equality holds (i.e., the energy of the input is equal to the energy of the output).

We can prove the energy-delay theorem as follows: Bessel’s equality gives

${\begin{aligned}\sum _{k=0}^{\infty }{y_{k}^{2}}={\frac {\Delta t}{2\pi }}\int \limits _{-\pi /\Delta t}^{\pi /\Delta t}{|}P\left(e^{i\omega \Delta t}\right)U\left(e^{i\omega \Delta t}\right){|}^{2}d\omega .\end{aligned}}$

(87)

Because $|P\left(e^{i\omega \Delta t}\right){|}^{2}=1$, the above equation becomes

${\begin{aligned}\sum _{k=0}^{\infty }{y_{k}^{2}}={\frac {\Delta t}{2\pi }}\int \limits _{-\pi /\Delta t}^{\pi /\Delta t}{|}U\left(e^{i\omega \Delta {\rm {t}}}\right){|}^{2}d\omega =\sum _{k=0}^{\infty }{u_{k}^{2}},\end{aligned}}$

(88)

which says that output and input have the same total energy. The truncated input $f_{k}$ (for time index n) is defined as

${\begin{aligned}f_{k}=u_{k}\mathrm {\;\;\;for\;} k\leq n,\;\;\;f_{k}=0\mathrm {\;\;\;for\;} k>n.\end{aligned}}$

(89)

In other words, the truncated input is the front end of the input signal. Let $g_{k}$ designate the output of the all-pass system to the truncated input $f_{k}$. Two statements can be made immediately about the output $g_{k}$: (1) because the system is all pass, the total energy of the truncated input $f_{k}$ is equal to the total energy of $g_{k}$; that is,

${\begin{aligned}\sum _{k=0}^{n}{u_{k}^{2}}=\sum _{k=0}^{n}{f_{k}^{2}}=\sum _{k=0}^{\infty }{g_{k}^{2}};\end{aligned}}$

(90)

and (2) because the system is causal, the front-end values of $y_{k}$ and $g_{k}$ are the same; that is,

${\begin{aligned}g_{k}=y_{k}\mathrm {\;\;for\;} k\leq n.\end{aligned}}$

(91)

Putting these two things together, we have

${\begin{aligned}\sum _{k=0}^{n}{u_{k}^{2}}=\sum _{k=0}^{\infty }{g_{k}^{2}}=\sum _{k=0}^{n}{y_{k}^{2}}+\sum _{k=n+1}^{\infty }{g_{k}^{2}}.\end{aligned}}$

(92)

Because the all-pass system is nontrivial, it can be shown that

${\begin{aligned}\sum _{k=n+1}^{\infty }{g_{k}^{2}}>0\end{aligned}}$

(93)

for some n. Hence, it follows that the front-end energy of the signal $u_{k}$ exceeds the front-end energy of the all-pass filtered signal $y_{k}$. In other words, an all-pass system delays the energy from input to output.

Let us consider a numerical example. The all-pass filter is $P\left(Z\right)=\left(0.5+Z\right)/\left({\rm {l\ +0.5}}Z\right)$, with impulse response $p_{k}$ (Table 1, first row below the column headings). The input signal to the all-pass filter is the wavelet $u_{k}$ (Table 1, second row) with three coefficients. The resulting output signal is the wavelet $y_{k}=u_{k}*p_{k}$ (Table 1, third row). For the cutoff n = 0, the front-end signal $f_{k}$ (i.e., the front end of the input $u_{k}$) is $f_{0}=u_{0}=-{\rm {0.75,}}{\rm {and}}{\rm {f}}_{\rm {k}}=0$ for other k (Table 1, fourth row). In other words, the front-end signal $f_{k}$ has only one nonzero coefficient, and it occurs at time 0. The resulting output of the all-pass system to the front-end signal is the signal $g_{k}$ (Table 1, fifth row). Note that the front ends of both the inputs u and f are the same (namely, –0.75), and the front ends of both the outputs y and g are the same (namely, –0.375).

Table 1. All-pass filter, input signal, resulting output, front end of input, and resulting output.

Time k	0	1	2	3	4
All-pass filter $p_{k}$	0.500	0.750	-0.375	0.188	-0.094
Input signal $u_{k}$	-0.750	2.750	1.000	0	0
Output $g_{k}$ for input signal	-0.375	0.813	2.844	-0.422	0.211
Front end $f_{k}$ of input signal	–0.750	0	0	0	0
Output $g_{k}$ for front end	-0.375	-0.563	0.282	-0.141	0.070

Table 2. Front end of $g_{k}$, tailgate of $g_{k}$, front-end energy of $g_{k}$, and components of tailgate energy of $g_{k}$.

Time k	0	1	2	3	4
Front end of $g_{k}$	–0.375	0	0	0	0
Tailgate of $g_{k}$	0	–0.563	0.282	–0.141	0.070
Front-end energy of $g_{k}$	${\left(-{0.375}\right)}^{2}$	0	0	0	0
Tailgate energy of $g_{k}$	0	$(-0.563)^{2}$	$(0.282)^{2}$	$(-0.141)^{2}$	$(0.070)^{2}$

The output $g_{k}$ is separated into two parts: (1) the front end (Table 2, first row below the column headings) and (2) the tailgate (Table 2, second row extended to infinity). The front-end energy of $g_{k}$ (Table 2, third row) is $g_{0}^{2}=y_{0}^{2}=\left(-{0.375}\right){=0.1406}$. The tailgate energy of $g_{k}$ is the sum of squares of its elements (Table 2, fourth row extended to infinity), which is 0.4219. Because the filter is an all-pass filter, the sum of front-end energy of $g_{k}$ and the tailgate energy of $g_{k}$, namely 0.1406 + 0.4219 = 0.5625, is equal to the input front-end energy, namely $f_{0}^{2}=u_{0}^{2}={\left(-{0.75}\right)}^{2}{=0.5625}$. Thus, the front-end energy $u_{0}^{2}{=0.5625}$ of the input to the all-pass filter exceeds the front-end energy $y_{0}^{2}{=0.l406}$ of the output. This demonstration is shown graphically in Figure 11.

Figure 11.  Graphic demonstration that the front-end energy of the input to an all-pass system exceeds that of the output.

The concept of minimum phase is attributable to Bode (1945). Let $a_{i}^{-1}$ be the poles and let $b_{i}^{-1}$ be the zeros of a system. A minimum-phase (or minimum-delay) system is a stable causal system that has its zeros as well as its poles lying outside of the unit circle. That is, stability requires that $|a_{i}|<1$, whereas minimum delay requires in addition that $|b_{i}|<1$. The canonical representation theorem (Robinson, 1963, theorem 1, p. 203) can be stated as follows:

The signal $g_{k}$ is stable and causal if and only if

${\begin{aligned}g_{k}=\sum _{n=0}^{\infty }{f_{n}}p_{k-n},\end{aligned}}$

(94)

where $f_{k}$ is the minimum-delay signal (with the same magnitude spectrum as that for $g_{k}$) and $p_{k}$ is an all-pass signal. In other words, a minimum-delay system is comparable to an executive who day by day receives written requests, acts on them as soon as possible, and immediately passes the results to a bureaucrat (an all-pass system). The bureaucrat holds the results, changes nothing, and then passes them on with various delays.

The inverse of the system $H\left(Z\right)$ is defined as the system 1/H(Z). Thus, the inverse of the ARMA(p,q) system B(Z)/A(Z) is the ARMA(q,p) system A(Z)/B(Z). The poles and zeros of the original system become, respectively, the zeros and poles of the inverse system. If $h_{k}$ is the impulse response for H(Z), then we write $h_{k}^{-1}$ for the impulse response for $H^{-1}\left(Z\right)=1/H\left(Z\right)$. However, $h_{k}^{-1}$ is not $1/h_{k}$, even as ${\rm {sin}}^{-1}\theta$ is not $1/\sin \theta$.

Because an AR(p) system is an all-pole system, it follows that a stable causal AR(p) system is necessarily a minimum-delay system. The inverse of a stable AR(p) system is an MA(p) system, in which the poles of the AR(p) system become the zeros of the MA(p) system. Thus we see that the inverse of an AR(p) system is a minimum-delay MA(p) system.

In our first such example, the given system is the stable causal AR(1) system $F\left(Z\right)={\left(1-aZ\right)}^{-1}$, where $|a|<1$. It has the stable causal impulse response $f_{k}=a^{k}$ for k = 0, 1, 2, …. The inverse system is the minimum-delay MA(1) system $F^{-1}\left(Z\right)=1-aZ$, with the impulse response $f_{k}^{-1}={\delta }_{k}-a{\delta }_{k-1}$.

In a second example, the given system is the minimum-delay ARMA(2,1) system

${\begin{aligned}F\left(Z\right)={\frac {{4}+Z}{\left({2}+Z\right)\left({3}+Z\right)}}={\frac {2}{{2}+Z}}-{\frac {1}{{3}+Z}},\end{aligned}}$

(95)

so the impulse response is the minimum-delay signal

${\begin{aligned}f_{k}={\left(-{\frac {1}{2}}\right)}^{k}-{\left(-{\frac {1}{3}}\right)}^{k}\mathrm {\;\;for\;\;} k=0,1,2,\ldots .\end{aligned}}$

(96)

The inverse system is the minimum-delay ARMA(1,2) system

${\begin{aligned}F^{-1}\left(Z\right)={\frac {\left({2}+Z\right)\left({3}+Z\right)}{{4}+Z}}=1+Z+{\frac {2}{{4}+Z}},\end{aligned}}$

(97)

so the impulse response of the inverse system is the causal signal

${\begin{aligned}f_{k}^{-1}={\delta }_{k}+{\delta }_{k-1}+{\frac {1}{2}}{\left(-{\frac {1}{4}}\right)}^{k}\mathrm {\;\;for\;\;} k=0,1,2,\ldots .\end{aligned}}$

(98)

In conclusion, let us state the (digital) minimum-delay theorem (Robinson, 1963, theorem 10, p. 205). Let the signal $f_{k}$ be a stable causal signal in the class of all stable causal signals with the same magnitude spectrum. Then each of the following conditions is necessary and sufficient for $f_{k}$ to be minimum delay:

1) The group delay of $f_{k}$ is minimum for all frequencies.

2) The front-end energy $\sum \limits _{k=0}^{n}{\;f_{k}^{2}}$ is a maximum for all time indices $n\geq 0$.

3) The inverse system $f_{k}^{-1}$ is causal.

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Also in this chapter

• Introduction
• Digital linear time-invariant systems
• Analog linear time-invariant systems
• Digital transfer functions
• Analog transfer functions
• Causality and stability of digital systems
• Causality and stability of analog systems
• Digital prediction
• Digital prediction error
• Analog prediction error

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