Discrete-Time Signal Transforms

This page documents discrete-time signal processing functions in Giac.jl.

Z-Transform

The Z-transform is a mathematical tool for analyzing discrete-time signals and systems. GIAC provides two commands for computing Z-transforms, available via Giac.Commands:

ztrans(expr, n, z) - Computes the unilateral Z-transform
invztrans(expr, z, n) - Computes the inverse Z-transform

Mathematical Definition

The unilateral Z-transform of a discrete sequence x[n] is defined as:

\[X(z) = \sum_{n=0}^{\infty} x[n] \cdot z^{-n}\]

Basic Usage

using Giac
using Giac.Commands: ztrans, invztrans

# Declare symbolic variables
@giac_var n z a

# Z-transform of a geometric sequence a^n
X = ztrans(a^n, n, z)
# Returns -z/(a-z) which is equivalent to z/(z-a)

# Z-transform of unit step (constant 1)
X_step = ztrans(1, n, z)
# Returns z/(z-1)

# Z-transform of ramp sequence n
X_ramp = ztrans(n, n, z)
# Returns z/(z-1)^2

Inverse Z-Transform

using Giac
using Giac.Commands: invztrans

@giac_var n z a

# Inverse Z-transform of z/(z-1) → unit step
x_step = invztrans(z/(z-1), z, n)
# Returns 1

# Inverse Z-transform of z/(z-a) → exponential
x_exp = invztrans(z/(z-a), z, n)
# Returns a^n

# Inverse Z-transform of z/(z-1)^2 → ramp
x_ramp = invztrans(z/(z-1)^2, z, n)
# Returns n

Round-Trip Verification

The Z-transform and inverse Z-transform are mathematical inverses:

using Giac
using Giac.Commands: ztrans, invztrans, simplify

@giac_var n z a

# Verify: invztrans(ztrans(a^n)) = a^n
original = a^n
transformed = ztrans(original, n, z)
recovered = invztrans(transformed, z, n)
simplified = simplify(recovered)
# Result: a^n

Common Z-Transform Pairs

Time Domain x[n]	Z-Domain X(z)	Region of Convergence
`1` (unit step)	`z/(z-1)`	\|z\| > 1
`n` (ramp)	`z/(z-1)²`	\|z\| > 1
`a^n`	`z/(z-a)`	\|z\| > \|a\|
`n·a^n`	`az/(z-a)²`	\|z\| > \|a\|

Using invoke_cmd

You can also use invoke_cmd directly:

using Giac

@giac_var n z a

# Using invoke_cmd
invoke_cmd(:ztrans, a^n, n, z)
invoke_cmd(:invztrans, z/(z-a), z, n)

Notes

Equivalent Forms: GIAC may return -z/(a-z) instead of z/(z-a). These are algebraically equivalent.
Variable Declaration: Always declare n and z as symbolic variables using @giac_var before using them in transforms.
Simplification: Use simplify from Giac.Commands to reduce results to canonical form.