Elementary Signals Quiz: Unit Step, Ramp, Impulse, and Exponential Quiz

Assess your understanding of elementary signals in signal processing, including unit step, ramp, impulse, and exponential signals. This quiz covers definitions, mathematical representations, and distinguishing features for a well-rounded review of these fundamental concepts.

1. Identifying the Unit Step Signal

   Which of the following best describes a unit step signal u(t) with respect to its value for t u003C 0 and t ≥ 0?

   1. It is 0 for t u003C 0 and 1 for t ≥ 0.
   2. It is -1 for t u003C 0 and 1 for t ≥ 0.
   3. It is t for all values of t.
   4. It is 1 for t u003C 0 and 0 for t ≥ 0.

   Explanation: The unit step signal u(t) is defined as zero for negative time and one for zero and positive time. The second option inverts this, which is incorrect; the third mixes negative and positive values, not characteristic of the step function. The fourth option describes a ramp rather than a step.

2. Ramp Signal Formation

   If you integrate the unit step signal u(t), which signal do you obtain as a result?

   1. Exponential signal e^{at}
   2. Sinusoidal signal sin(t)
   3. Unit ramp signal r(t)
   4. Unit impulse signal δ(t)

   Explanation: Integrating the unit step signal produces the unit ramp signal r(t). The impulse signal is related to differentiation, not integration; the exponential signal arises from a different functional form; and the sinusoidal signal is unrelated to the step signal in this context.

3. Impulse Signal Properties

   Which feature uniquely characterizes the unit impulse signal δ(t) among elementary signals?

   1. It increases linearly for all t.
   2. It decays exponentially for all t.
   3. It is zero everywhere except at t = 0 and integrates to one.
   4. It is one for all t ≥ 0.

   Explanation: The unit impulse δ(t) is nonzero only at the origin and its area under the curve is one, which is unique among elementary signals. The second option describes the unit step, the third refers to the ramp, and the fourth to the exponential decay, none of which share the unique properties of the impulse.

4. Exponential Signal Classification

   For the exponential signal x(t) = e^{-2t}, what behavior is observed as t increases?

   1. The signal grows without bound.
   2. The signal oscillates between plus and minus one.
   3. The signal remains constant at one.
   4. The signal decays rapidly toward zero.

   Explanation: With a negative exponent, the exponential signal decays rapidly to zero as t increases. Growth without bound would occur with a positive exponent; a constant value arises only if the exponent is zero; oscillation would indicate a sinusoidal, not exponential, function.

5. Relationship Between Impulse and Step Signals

   Which mathematical operation on the unit step signal u(t) results in the unit impulse signal δ(t)?

   1. Addition of a constant
   2. Differentiation
   3. Integration
   4. Multiplication by t

   Explanation: Differentiating the unit step signal yields the unit impulse signal. Integration would produce a ramp, multiplication by t would result in a polynomial, and adding a constant alters only the signal’s offset, not its fundamental type.