Sampling and Z-Transform (Step-by-Step)

Step 1: Given Signal

Continuous-time signal:

x(t) = e^-t

Step 2: Sampling

Sampling frequency:

f_s = 10 Hz

Sampling period:

T = 1 / f_s = 0.1 sec

Discrete signal:

x[n] = x(nT) = e^-0.1n

Step 3: Z-Transform

Definition:

X(z) = Σ x[n] z^-n

Substitute x[n]:

X(z) = Σ (e^-0.1)ⁿ z^-n

This is a geometric series:

X(z) = 1 / (1 - e^-0.1 z^-1)

Multiply numerator and denominator by z:

X(z) = z / (z - e^-0.1)

Step 4: Numerical Value

Compute:

e^-0.1 ≈ 0.9048

Final answer:

X(z) = z / (z - 0.9048)

Step 5: Special Case

If:

x[n] = (0.5)ⁿ

Then:

X(z) = z / (z - 0.5)

This corresponds to a different continuous signal:

x(t) = e^-6.93t

Conclusion

• Sampling converts continuous signal to discrete signal.
• Z-transform is used for sampled signals.
• Always compute e^-aT carefully.