Referencia: Tabla de transformadas z: Lathi Tabla 5.1 Transformada z p492. Oppenheim tabla 10.2 p776, Schaum Hsu Tabla 4-1 p170. Propiedades: Schaum Hsu Tabla 4-2 p173. Lathi Tabla 5.2 p509. Oppenheim Tabla 10.3 p793
Usando los algoritmos presentados para transformadas z a partir de la tabla y propiedades, se incorpora en la instrucción z_transform(f,n,z) incorporada en telg1001.py. Para demostración se desarrollan algunos ejemplos que se desarrollan a partir de la tabla de transformadas z.
Transformada z de impulso unitario δ(t)
>>> import sympy as sym
>>> import telg1001_z as fcnm
>>> n = sym.Symbol('n', real=True)
>>> z = sym.Symbol('z')
>>> d = sym.DiracDelta(n)
>>> f = d
>>> fcnm.z_transform(f,n,z)
1
>>>
Transformada z de impulso unitario desplazado δ(t-2) y δ(t+2)
>>> fcnm.z_transform(f.subs(n,n-2),n,z) z**(-2) >>> fcnm.z_transform(f.subs(n,n+2),n,z) z**2 >>>
Tabla de Transformadas z unilateral e Inversas usando algoritmo en telg1001.py
Para comprobar las transformadas de los algoritmos presentados e integrados en telg1001.py, se crea una tabla con expresiones f[n] de prueba en el dominio n de tiempo discreto.
Al resultado se aplica la inversa de F[z] que muestra que la expresión es invertible usando la transformada z unilateral.
El resultado del algoritmo se muestra a continuación y luego el algoritmo.
z_transform : 0 f[n]: DiracDelta(n) F[z]: (1, True, True) f[n]: (DiracDelta(n), True, True) z_transform : 1 f[n]: DiracDelta(n - 2) F[z]: (z**(-2), Abs(z) > 0, True) f[n]: (DiracDelta(n - 2), True, True) z_transform : 2 f[n]: DiracDelta(n + 2) F[z]: (z**2, True, True) f[n]: (DiracDelta(n + 2), True, True) z_transform : 3 f[n]: 3*DiracDelta(n + 2) F[z]: (3*z**2, True, True) f[n]: (3*DiracDelta(n + 2), True, True) z_transform : 4 f[n]: -3*DiracDelta(n + 2) F[z]: (-3*z**2, True, True) f[n]: (-3*DiracDelta(n + 2), True, True) z_transform : 5 f[n]: Heaviside(n) F[z]: (z/(z - 1), Abs(z) > 1, True) f[n]: (Heaviside(n), Abs(z) > 1, True) z_transform : 6 f[n]: Heaviside(n - 2) F[z]: (1/(z*(z - 1)), Abs(z) > 1, True) f[n]: [None, [1/(z*(z - 1))]] z_transform : 7 f[n]: Heaviside(n + 2) F[z]: (z**3/(z - 1), Abs(z) > 1, True) f[n]: (Heaviside(n + 2), Abs(z) > 1, True) z_transform : 8 f[n]: 3*Heaviside(n + 2) F[z]: (3*z**3/(z - 1), Abs(z) > 1, True) f[n]: (3*Heaviside(n + 2), Abs(z) > 1, True) z_transform : 9 f[n]: n*Heaviside(n) F[z]: (z/(z - 1)**2, Abs(z) > 1, True) f[n]: (n*Heaviside(n), Abs(z) > 1, True) z_transform : 10 f[n]: n**2*Heaviside(n) F[z]: (z*(z + 1)/(z - 1)**3, Abs(z) > 1, True) f[n]: (n**2*Heaviside(n), Abs(z) > 1, True) z_transform : 11 f[n]: n**3*Heaviside(n) F[z]: (z*(z**2 + 4*z + 1)/(z - 1)**4, Abs(z) > 1, True) f[n]: (n**3*Heaviside(n), Abs(z) > 1, True) z_transform : 12 f[n]: 0.5**n*Heaviside(n) F[z]: (z/(z - 0.5), Abs(z) > 0.5, True) f[n]: (0.5**n*Heaviside(n), Abs(z) > 0.5, True) z_transform : 13 f[n]: 0.5**n*Heaviside(n) F[z]: (z/(z - 0.5), Abs(z) > 0.5, True) f[n]: (0.5**n*Heaviside(n), Abs(z) > 0.5, True) z_transform : 14 f[n]: Heaviside(n)/4**n F[z]: (z/(z - 1/4), Abs(z) > 1/4, True) f[n]: (Heaviside(n)/4**n, Abs(z) > 1/4, True) z_transform : 15 f[n]: -0.5**n*Heaviside(-n - 1) F[z]: (z/(z - 0.5), Abs(z) < 0.5, True) f[n]: (0.5**n*Heaviside(n), Abs(z) > 0.5, True) z_transform : 16 f[n]: 0.5**(n - 1)*Heaviside(n - 1) F[z]: (1.0/(z - 0.5), Abs(z) > 0.5, True) f[n]: [None, [1/(z - 0.5)]] z_transform : 17 f[n]: 4**(1 - n)*Heaviside(n - 1) F[z]: (1/(z - 1/4), Abs(z) > 1/4, True) f[n]: [None, [1/(z - 1/4)]] z_transform : 18 f[n]: 0.5**n*n*Heaviside(n) F[z]: (0.5*z/(z - 0.5)**2, Abs(z) > 0.5, True) f[n]: (0.5**n*n*Heaviside(n), Abs(z) > 0.5, True) z_transform : 19 f[n]: -0.5**n*n*Heaviside(-n - 1) F[z]: (0.5*z/(z - 0.5)**2, Abs(z) < 0.5, True) f[n]: (0.5**n*n*Heaviside(n), Abs(z) > 0.5, True) z_transform : 20 f[n]: 0.5**n*(n + 1)*Heaviside(n) F[z]: z/(z - 0.5) + 0.5*z/(z - 0.5)**2 f[n]: (2*0.5**(n + 1)*(n + 1)*Heaviside(n + 1), Abs(z) > 0.5, True) z_transform : 21 f[n]: 0.5**n*n**2*Heaviside(n) F[z]: (0.5*z*(z + 0.5)/(z - 0.5)**3, Abs(z) > 0.5, True) f[n]: (0.5**n*n**2*Heaviside(n), Abs(z) > 0.5, True) z_transform : 22 f[n]: 1.33333333333333*0.5**n*n*(n - 2)*(n - 1)*Heaviside(n) F[z]: 1.33333333333333*z/(z - 0.5)**2 - 2*z*(z + 0.5)/(z - 0.5)**3 + 0.666666666666667*z*(z**2 + 2*z + 0.25)/(z - 0.5)**4 f[n]: (1.33333333333333*0.5**n*n*(n - 2)*(n - 1)*Heaviside(n), Abs(z) > 1, 0) z_transform : 23 f[n]: cos(2*n)*Heaviside(n) F[z]: (z*(z - cos(2))/(z**2 - 2*z*cos(2) + 1), Abs(z) > 1, True) f[n]: (cos(2*n)*Heaviside(n), Abs(z) > 1, True) z_transform : 24 f[n]: 0.5**n*cos(2*n)*Heaviside(n) F[z]: (z*(z - 0.5*cos(2))/(z**2 - z*cos(2) + 0.25), Abs(z) > 0.5, True) f[n]: (0.5**n*cos(2*n)*Heaviside(n), Abs(z) > 0.5, True) z_transform : 25 f[n]: sin(2*n)*Heaviside(n) F[z]: (z*sin(2)/(z**2 - 2*z*cos(2) + 1), Abs(z) > 1, True) f[n]: (sin(2*n)*Heaviside(n), Abs(z) > 1, True) z_transform : 26 f[n]: 0.5**n*sin(2*n)*Heaviside(n) F[z]: (0.5*z*sin(2)/(z**2 - z*cos(2) + 0.25), Abs(z) > 0.5, True) f[n]: (0.5**n*sin(2*n)*Heaviside(n), Abs(z) > 0.5, True) z_transform : 27 f[n]: cos(2*n + 0.25)*Heaviside(n) F[z]: (z*(z*cos(0.25) - cos(1.75))/(z**2 - 2*z*cos(2) + 1), Abs(z) > 1, True) f[n]: (1.0*cos(2*n + 0.25)*Heaviside(n), Abs(z) > 1, True) z_transform : 28 f[n]: 3*0.5**n*cos(2*n + 0.25)*Heaviside(n) F[z]: (3*z*(z*cos(0.25) - 0.5*cos(1.75))/(z**2 - z*cos(2) + 0.25), Abs(z) > 0.5, True) f[n]: (3.0*0.5**n*cos(2*n + 0.25)*Heaviside(n), Abs(z) > 0.5, True)
Algoritmo en Python
Las intrucciones se pueden modificar para observar una expresión en detalle si se activa la variable sym.SYMPY_DEBUG=True. Por la extensión del resultado se recomienda activarla para una expresión.
import sympy as sym import telg1001 as fcnm sym.SYMPY_DEBUG = False # INGRESO n = sym.Symbol('n') z = sym.Symbol('z') u = sym.Heaviside(n) d = sym.DiracDelta(n) a0 = sym.Rational(1,4).limit_denominator(100) tabla = [ d, d.subs(n,n-2), d.subs(n,n+2), 3*d.subs(n,n+2), -3*d.subs(n,n+2), u, u.subs(n,n-2), u.subs(n,n+2), 3*u.subs(n,n+2), n*u, (n**2)*u, (n**3)*u, 0.5**n*u,((1/2)**n)*u,(a0**n)*u, -(0.5**n)*u.subs(n,-n-1), (0.5**(n-1))*u.subs(n,n-1), (a0**(n-1))*u.subs(n,n-1), n*(0.5**n)*u, -n*(0.5**n)*u.subs(n,-n-1), (n+1)*(0.5**n)*u, n**2*(0.5**n)*u, n*(n-1)*(n-2)*(0.5**n)*u/((0.5**3)*6), sym.cos(2*n)*u, (0.5**n)*sym.cos(2*n)*u, sym.sin(2*n)*u, (0.5**n)*sym.sin(2*n)*u, sym.cos(2*n+0.25)*u, 3*(0.5**n)*sym.cos(2*n+0.25)*u ] # seleccionar revisar revisar = tabla[:] # revisa detalle de una transformada y la inversa if len(revisar)<2: sym.SYMPY_DEBUG = True else: sym.SYMPY_DEBUG = False # PROCEDIMIENTO for i in range(0,len(revisar),1): f = revisar[i] # SALIDA print('\nz_transform : ', i) print(' f[n]:',f) Fz = fcnm.z_transform(f,n,z) print(' F[z]:',Fz) if type(Fz)==tuple: fn = fcnm.inverse_z_transform(Fz[0],z,n) else: fn = fcnm.inverse_z_transform(Fz,z,n) print(' f[n]:',fn)