--- title: "InverseBilateralZTransform" language: "en" type: "Symbol" summary: "InverseBilateralZTransform[expr, z, n] gives the inverse bilateral Z transform of expr. InverseBilateralZTransform[expr, {z1, ..., zk}, {n1, ..., nk}] gives the multidimensional inverse bilateral Z transform of expr." keywords: - generating function - impulse response - inverse bilateral Z-transform - inverse bilateral z transform - ramp response - series coefficient - step response - transfer function - bilateral z transform canonical_url: "https://reference.wolfram.com/language/ref/InverseBilateralZTransform.html" source: "Wolfram Language Documentation" related_guides: - title: "Summation Transforms" link: "https://reference.wolfram.com/language/guide/SummationTransforms.en.md" related_functions: - title: "BilateralZTransform" link: "https://reference.wolfram.com/language/ref/BilateralZTransform.en.md" - title: "ZTransform" link: "https://reference.wolfram.com/language/ref/ZTransform.en.md" - title: "InverseZTransform" link: "https://reference.wolfram.com/language/ref/InverseZTransform.en.md" - title: "InverseBilateralLaplaceTransform" link: "https://reference.wolfram.com/language/ref/InverseBilateralLaplaceTransform.en.md" - title: "GeneratingFunction" link: "https://reference.wolfram.com/language/ref/GeneratingFunction.en.md" - title: "SeriesCoefficient" link: "https://reference.wolfram.com/language/ref/SeriesCoefficient.en.md" - title: "InverseFourierSequenceTransform" link: "https://reference.wolfram.com/language/ref/InverseFourierSequenceTransform.en.md" --- # InverseBilateralZTransform InverseBilateralZTransform[expr, z, n] gives the inverse bilateral Z transform of expr. InverseBilateralZTransform[expr, {z1, …, zk}, {n1, …, nk}] gives the multidimensional inverse bilateral Z transform of expr. ## Details and Options * The inverse bilateral Z transform provides the map from Fourier space back to state space, and allows one to recover the original sequence in applications of the bilateral Z transform. * The inverse bilateral Z transform of a function $F(z)$ is given by the contour integral $\frac{1}{2\pi i}\oint _CF(z)z^{n-1}dz$, where the integration is along a counterclockwise contour $\gamma$, lying in an annulus $\alpha <| z| <\beta$ in which the function $F(z)$ is holomorphic. In some cases, the annulus of analyticity may extend to the interior or the exterior of a disk. [image] * The multidimensional inverse transform is given by $\frac{1}{(2 \pi i)^m}\oint _CF\left(z_1,\ldots ,z_k\right)z_1{}^{n_1-1}\ldots z_k{}^{n_k-1}dz_1\ldots dz_k$, where $m=n_1+\ldots +n_k$. * The following options can be given: | | | | | ------------------ | ----------------- | ------------------------------------------------------------------ | | AccuracyGoal | Automatic | digits of absolute accuracy sought | | Assumptions | \$Assumptions | assumptions to make about parameters | | GenerateConditions | False | whether to generate answers that involve conditions on parameters | | Method | Automatic | method to use | | PerformanceGoal | \$PerformanceGoal | aspects of performance to optimize | | PrecisionGoal | Automatic | digits of precision sought | | WorkingPrecision | Automatic | the precision used in internal computations | --- ## Examples (22) ### Basic Examples (5) Inverse bilateral Z transform of a rational function defined in an annulus: ```wl In[1]:= F = ConditionalExpression[(z/(z - 1)(z - 3)), 1 < Abs[z] < 3] Out[1]= ConditionalExpression[z/((-3 + z)*(-1 + z)), 1 < Abs[z] < 3] In[2]:= ComplexRegionPlot[1 < Abs[z] < 3, ...] Out[2]= [image] In[3]:= InverseBilateralZTransform[F, z, n] Out[3]= Piecewise[{{-(1/2), n > 0}}, -(3^n/2)] In[4]:= DiscretePlot[%, {n, -5, 5}, PlotRange -> All] Out[4]= [image] ``` Compute the inverse transform at a single point: ```wl In[5]:= InverseBilateralZTransform[F, z, -5, WorkingPrecision -> MachinePrecision] Out[5]= -0.00205761 ``` --- Function defined in the exterior of a disk: ```wl In[1]:= InverseBilateralZTransform[ConditionalExpression[(1/z - 1), Abs[z] > 1], z, n] Out[1]= Piecewise[{{1, n > 0}}, 0] In[2]:= DiscretePlot[%, {n, -5, 5}, PlotRange -> All] Out[2]= [image] ``` --- Function defined in the interior of a disk: ```wl In[1]:= InverseBilateralZTransform[ConditionalExpression[(1/z - 1), Abs[z] < 1], z, n] Out[1]= Piecewise[{{-1, n <= 0}}, 0] In[2]:= DiscretePlot[%, {n, -5, 5}, PlotRange -> All] Out[2]= [image] ``` --- Function with an essential singularity at zero: ```wl In[1]:= F = Exp[1 / z] Out[1]= E^(1/z) In[2]:= ComplexPlot[F, {z, -1 - I, +1 + 1I}, Rule[...]] Out[2]= [image] In[3]:= InverseBilateralZTransform[F, z, n] Out[3]= (UnitStep[n]/n!) In[4]:= DiscretePlot[%, {n, -5, 5}, PlotRange -> All] Out[4]= [image] ``` --- Multivariate inverse bilateral transform: ```wl In[1]:= InverseBilateralZTransform[ConditionalExpression[(z w/(z + 1)^2(w + 1)^2), Abs[z] > 1 && Abs[w] > 1], {z, w}, {n, m}] Out[1]= Piecewise[{{(-1)^(m + n)*m*n, m >= 0 && n >= 0}}, 0] ``` ### Scope (7) Shifted impulse sequences: ```wl In[1]:= InverseBilateralZTransform[ConditionalExpression[z^5, Abs[z] < Infinity], z, n] Out[1]= Piecewise[{{1, n == -5}}, 0] In[2]:= InverseBilateralZTransform[ConditionalExpression[z^-5, 0 < Abs[z]], z, n] Out[2]= Piecewise[{{1, n == 5}}, 0] ``` --- Rational functions yield exponential and trigonometric sequences: ```wl In[1]:= F = ConditionalExpression[(z^4/(z ^ 2 + 1)), Abs[z] < 1] Out[1]= ConditionalExpression[z^4/(1 + z^2), Abs[z] < 1] In[2]:= ComplexPlot[F[[1]], {z, -2 - 2I, +2 + 2I}, Rule[...]] Out[2]= [image] In[3]:= x[n_] = InverseBilateralZTransform[F, z, n]//FullSimplify Out[3]= Piecewise[{{Cos[(n*Pi)/2], n == -1 || n < -2}}, 0] In[4]:= DiscretePlot[x[n], {n, -20, 5}] Out[4]= [image] ``` --- Functions involving parameters: ```wl In[1]:= InverseBilateralZTransform[ConditionalExpression[(1/(z - a)(z - b)), a < Abs[z] < b], z, n] Out[1]= Piecewise[{{a^(-1 + n)/(a - b), n > 0}}, b^(-1 + n)/(a - b)] ``` --- The following function defined in the interior of a circle leads to a trigonometric sequence: ```wl In[1]:= InverseBilateralZTransform[ConditionalExpression[(1 - Cos[ω]z^-1/(1 - 2Cos[ω]z^-1 + z^-2)), Abs[z] < 1], z, n] Out[1]= Piecewise[{{(1/2)*(ChebyshevU[-2 - n, Cos[ω]] - ChebyshevU[-n, Cos[ω]]), n <= -2 || n == -1}}, 0] In[2]:= FullSimplify[%, n∈Integers] Out[2]= Piecewise[{{-Cos[n*ω], n < 0}}, 0] ``` --- If the ROC is not provided, then it is assumed to be the region containing all the function poles: ```wl In[1]:= InverseBilateralZTransform[(z/(z - 1)(z ^ 2 + 1)), z, n]//FullSimplify Out[1]= -(1/2) (-1 + Cos[(n π/2)] + Sin[(n π/2)]) UnitStep[n] ``` Obtain the same result using ``InverseZTransform`` : ```wl In[2]:= InverseZTransform[(z/(z - 1)(z ^ 2 + 1)), z, n]//FullSimplify Out[2]= (1/2) (1 - Cos[(n π/2)] - Sin[(n π/2)]) ``` --- Calculate the inverse bilateral Z transform at a single point using a numerical method: ```wl In[1]:= InverseBilateralZTransform[ConditionalExpression[(z/(z^2 + 4)(z^2 + 25)), 2 < Abs[z] < 5], z, 11, WorkingPrecision -> MachinePrecision] Out[1]= -48.7619 ``` Alternatively, calculate inverse symbolically: ```wl In[2]:= InverseBilateralZTransform[ConditionalExpression[(z/(z^2 + 4)(z^2 + 25)), 2 < Abs[z] < 5], z, n] Out[2]= Piecewise[{{(1/84)*I*((-2*I)^n - (2*I)^n), n > 0}}, (1/210)*I*((-5*I)^n - (5*I)^n)] ``` Then evaluate it for a specific value of $n$ : ```wl In[3]:= N[% /. n -> 11] Out[3]= -48.7619 ``` --- For some functions, the inverse bilateral Z transform can be evaluated only numerically: ```wl In[1]:= f[n_ ? NumericQ] := InverseBilateralZTransform[ConditionalExpression[(10 2^Sin[z^2]/((z ^ 2 + 1)(z - 3))), 1 < Abs[z] < 3], z, n, WorkingPrecision -> MachinePrecision] In[2]:= f[5] Out[2]= -0.558074 - 7.942780652783198`*^-10 I ``` Plot the inverse bilateral Z transform using numerical values only: ```wl In[3]:= DiscretePlot[Re[f[n]], {n, -10, 10}] Out[3]= [image] ``` ### Options (2) #### Assumptions (1) Use ``Assumptions`` to restrict the parameter domain: ```wl In[1]:= InverseBilateralZTransform[ConditionalExpression[(z ^ 2 + 1/(z - 1)(z - a)), 1 < Abs[z] < 3], z, n, Assumptions -> a ≥ 3] Out[1]= -(2/-1 + a) + (Piecewise[{{1/a, n == 0}, {-((-2*a + a^n + a^(2 + n))/((-1 + a)*a)), n < 0}}, 0]) ``` #### WorkingPrecision (1) Use ``WorkingPrecision`` to obtain a result with arbitrary precision: ```wl In[1]:= F = ConditionalExpression[Exp[z ^ 2 / (z - 1)] * Sin[z], Abs[z] < 1]; In[2]:= InverseBilateralZTransform[F, z, -7, WorkingPrecision -> MachinePrecision] Out[2]= 0.408135 In[3]:= InverseBilateralZTransform[F, z, -7, WorkingPrecision -> 10] Out[3]= 0.4081349206 In[4]:= InverseBilateralZTransform[F, z, -7, WorkingPrecision -> 20] Out[4]= 0.40813492063492063492 ``` ### Applications (2) Define finite duration and exponentially decaying signals: ```wl In[1]:= x1[n_] := 3 / 4(DiscreteDelta[n + 1] + DiscreteDelta[n] + DiscreteDelta[n - 1]) x2[n_] := 2^-Abs[n] ``` Plot signals in the time domain: ```wl In[2]:= DiscretePlot[{x1[n], x2[n]}, {n, -5, 5}, PlotRange -> All] Out[2]= [image] ``` To find the convolution, first calculate product of the transforms: ```wl In[3]:= BilateralZTransform[x1[n], n, z]BilateralZTransform[x2[n], n, z] Out[3]= ConditionalExpression[-((9*z*(1 + 1/z + z))/(4*(2 - 5*z + 2*z^2))), 1/2 < Abs[z] < 2] ``` Then, perform inversion back to the time domain: ```wl In[4]:= InverseBilateralZTransform[%, z, n]//FullSimplify Out[4]= Piecewise[{{21*2^(-3 - n), n > 0}, {21*2^(-3 + n), n < 0}}, 3/2] ``` Plot the convolution in the time domain: ```wl In[5]:= DiscretePlot[%, {n, -5, 5}] Out[5]= [image] ``` Alternatively, find the convolution using ``DiscreteConvolve`` : ```wl In[6]:= DiscreteConvolve[x1[k], x2[k], k, n]//FullSimplify Out[6]= (3/4) (2^-Abs[-1 + n] + 2^-Abs[n] + 2^-Abs[1 + n]) ``` --- Define a pair of infinite duration signals: ```wl In[1]:= x1[n_] := n UnitStep[n] x2[n_] := 2^nUnitStep[n - 1] ``` Plot the signals in the time domain: ```wl In[2]:= DiscretePlot[{x1[n], x2[n]}, {n, 0, 7}] Out[2]= [image] ``` To find the convolution, first calculate the product of the transforms: ```wl In[3]:= BilateralZTransform[x1[n], n, z]BilateralZTransform[x2[n], n, z] Out[3]= ConditionalExpression[(2*z)/((-2 + z)*(-1 + z)^2), Abs[z] > 2] ``` Perform inversion back to the time domain: ```wl In[4]:= InverseBilateralZTransform[%, z, n] Out[4]= Piecewise[{{2*(-1 + 2^n - n), n >= 0}}, 0] ``` Plot the convolution in the time domain: ```wl In[5]:= DiscretePlot[%, {n, 0, 6}] Out[5]= [image] ``` Alternatively, find the convolution using ``DiscreteConvolve`` : ```wl In[6]:= DiscreteConvolve[x1[k], x2[k], k, n] Out[6]= Piecewise[{{2*(-1 + 2^n - n), n > 1}}, 0] ``` ### Properties & Relations (4) Relation to ``BilateralZTransform`` : ```wl In[1]:= InverseBilateralZTransform[BilateralZTransform[x[n], n, z], z, n] Out[1]= x[n] ``` --- ``InverseBilateralZTransform`` is closely related to ``InverseFourierSequenceTransform`` : ```wl In[1]:= {InverseBilateralZTransform[ConditionalExpression[(z^2/1 / 3 + z), Abs[z] > 1 / 3], z, n], InverseFourierSequenceTransform[(z^2/1 / 3 + z) /. z -> Exp[I ω], ω, n]//Simplify} Out[1]= {Piecewise[{{(-(1/3))^(1 + n), n == -1 || n >= 0}}, 0], Piecewise[{{(-(1/3))^(1 + n), n == -1 || n >= 0}}, 0]} In[2]:= {InverseBilateralZTransform[z^-10, z, n], InverseFourierSequenceTransform[z^-10 /. z -> Exp[I ω], ω, n]} Out[2]= {Piecewise[{{1, n == 10}}, 0], DiscreteDelta[-10 + n]} ``` --- Linearity: ```wl In[1]:= InverseBilateralZTransform[a F[z] + b G[z], z, n] Out[1]= a InverseBilateralZTransform[F[z], z, n] + b InverseBilateralZTransform[G[z], z, n] ``` --- Scaling: ```wl In[1]:= InverseBilateralZTransform[ F[a * z], z, n] Out[1]= a^-n InverseBilateralZTransform[F[z], z, n] ``` ### Possible Issues (1) No poles are allowed in the region of convergence: ```wl In[1]:= InverseBilateralZTransform[ConditionalExpression[(z/(z - 1)(z - 3)), 1 < Abs[z] < 4], z, n] Out[1]= InverseBilateralZTransform[ConditionalExpression[z/(3 - 4*z + z^2), 1 < Abs[z] < 4], z, n] ``` ### Neat Examples (1) Create a table of basic inverse bilateral Z transforms: ```wl In[1]:= CompoundExpression[...] In[2]:= Grid[Prepend[{#, FullSimplify[InverseBilateralZTransform[#1, z, n]]}& /@ Flist, {F[z], InverseBilateralZTransform[F[z], z, n]}], IconizedObject[«Grid options»]]//TraditionalForm Out[2]//TraditionalForm= $$\begin{array}{cc} F(z) & \text{InverseBilateralZTransform}[F(z),z,n] \\ \fbox{$\frac{1}{z}\text{ if }| z| >0$} & \begin{array}{ll} \{ & \begin{array}{ll} 1 & n=1 \\ 0 & \text{True} \\ \end{array} \\ \end{array} \\ \fbox{$z\text{ if }| z ... 4\right)}\text{ if }\frac{1}{2}<| z| <2$} & \begin{array}{ll} \{ & \begin{array}{ll} 2^{2-n} \cos \left(\frac{\pi n}{2}\right) & n>0 \\ 2^{n-2} \cos \left(\frac{\pi n}{2}\right) & \text{True} \\ \end{array} \\ \end{array} \\ \end{array}$$ ``` ## See Also * [`BilateralZTransform`](https://reference.wolfram.com/language/ref/BilateralZTransform.en.md) * [`ZTransform`](https://reference.wolfram.com/language/ref/ZTransform.en.md) * [`InverseZTransform`](https://reference.wolfram.com/language/ref/InverseZTransform.en.md) * [`InverseBilateralLaplaceTransform`](https://reference.wolfram.com/language/ref/InverseBilateralLaplaceTransform.en.md) * [`GeneratingFunction`](https://reference.wolfram.com/language/ref/GeneratingFunction.en.md) * [`SeriesCoefficient`](https://reference.wolfram.com/language/ref/SeriesCoefficient.en.md) * [`InverseFourierSequenceTransform`](https://reference.wolfram.com/language/ref/InverseFourierSequenceTransform.en.md) ## Related Guides * [Summation Transforms](https://reference.wolfram.com/language/guide/SummationTransforms.en.md) ## History * [Introduced in 2021 (13.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn130.en.md)