সিস্টেম — লিনিয়ারিটি, টাইম-ইনভ্যারিয়েন্স ও LTI সিস্টেম

x1 = [1.0, 2.0, 3.0, 4.0]
x2 = [0.5, -1.0, 2.0, 0.0]
a, b = 2.0, 3.0

def system_linear(x):
    return [2.0 * v for v in x]        # y[n] = 2*x[n]

def system_nonlinear(x):
    return [v**2 for v in x]           # y[n] = x[n]^2

def combine(a, xa, b, xb):
    return [a*xa[n] + b*xb[n] for n in range(len(xa))]

x_comb = combine(a, x1, b, x2)         # a*x1 + b*x2 (ইনপুট মেশানো)
print(f"x1={x1}\nx2={x2}\na={a}, b={b}")
print(f"a*x1+b*x2 = {x_comb}\n")

print("-- System 1: y[n] = 2*x[n] --")
lhs1 = system_linear(x_comb)                                              # T(a*x1+b*x2)
rhs1 = [a*v1 + b*v2 for v1, v2 in zip(system_linear(x1), system_linear(x2))]  # a*T(x1)+b*T(x2)
print(f"T(a*x1+b*x2)      = {lhs1}")
print(f"a*T(x1)+b*T(x2)   = {rhs1}")
match1 = all(abs(lhs1[n]-rhs1[n]) < 1e-9 for n in range(len(lhs1)))
print(f"match: {match1} -> {'LINEAR' if match1 else 'NOT LINEAR'}\n")

print("-- System 2: y[n] = x[n]^2 --")
lhs2 = system_nonlinear(x_comb)
rhs2 = [a*v1 + b*v2 for v1, v2 in zip(system_nonlinear(x1), system_nonlinear(x2))]
print(f"T(a*x1+b*x2)      = {lhs2}")
print(f"a*T(x1)+b*T(x2)   = {rhs2}")
match2 = all(abs(lhs2[n]-rhs2[n]) < 1e-9 for n in range(len(lhs2)))
print(f"match: {match2} -> {'LINEAR' if match2 else 'NOT LINEAR (সুপারপজিশন ফেল)'}")
print(f"per-sample পার্থক্য: {[round(abs(lhs2[n]-rhs2[n]),4) for n in range(len(lhs2))]}")

x3 = [1.0, 2.0, 3.0, 4.0, 5.0, 0.0]
shift = 2

def shift_signal(x, k):
    return [0.0]*k + x[:len(x)-k]      # ডানে k স্যাম্পল শিফট, বামে জিরো-প্যাড

def system_avg(x):
    y = []
    for n in range(len(x)):
        prev = x[n-1] if n-1 >= 0 else 0.0
        y.append(0.5*x[n] + 0.5*prev)  # y[n] = 0.5*x[n] + 0.5*x[n-1]
    return y

def system_time_varying(x):
    return [n * x[n] for n in range(len(x))]   # y[n] = n*x[n] -- সরাসরি n-এর উপর নির্ভরশীল

print(f"x[n] = {x3}, shift k={shift}")
x3_shifted = shift_signal(x3, shift)
print(f"x[n-{shift}] = {x3_shifted}\n")

print("-- System: y[n] = 0.5*x[n] + 0.5*x[n-1] --")
y_then_shift = shift_signal(system_avg(x3), shift)     # shift(T(x))
y_shift_then = system_avg(x3_shifted)                   # T(shifted x)
print(f"shift(T(x[n]))  = {y_then_shift}")
print(f"T(x[n-{shift}])      = {y_shift_then}")
match3 = all(abs(y_then_shift[n]-y_shift_then[n]) < 1e-9 for n in range(len(x3)))
print(f"match: {match3} -> {'TIME-INVARIANT' if match3 else 'NOT time-invariant'}\n")

print("-- System: y[n] = n * x[n] --")
y_then_shift2 = shift_signal(system_time_varying(x3), shift)
y_shift_then2 = system_time_varying(x3_shifted)
print(f"shift(T(x[n]))  = {y_then_shift2}")
print(f"T(x[n-{shift}])      = {y_shift_then2}")
match4 = all(abs(y_then_shift2[n]-y_shift_then2[n]) < 1e-9 for n in range(len(x3)))
print(f"match: {match4} -> {'TIME-INVARIANT' if match4 else 'NOT time-invariant'}")
print(f"per-sample পার্থক্য: {[round(abs(y_then_shift2[n]-y_shift_then2[n]),4) for n in range(len(x3))]}")