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Quiz: Introduction to Signals and Systems

Test your understanding of fundamental signal and system concepts.

1. What is a system in the context of signal processing?

  1. A collection of unrelated mathematical equations
  2. An entity that processes input signals to produce output signals according to specific rules
  3. A physical device only, not including software algorithms
  4. A method for storing digital data
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The correct answer is B. A system is an entity that processes input signals to produce output signals according to specific rules or mathematical operations. The fundamental relationship is expressed as \(y(t) = \mathcal{T}[x(t)]\), where \(x(t)\) is the input signal, \(y(t)\) is the output signal, and \(\mathcal{T}\) represents the transformation operation. Systems can be physical devices like filters or amplifiers, or algorithmic processes implemented in software.

Concept Tested: Systems

See: Systems

2. Which notation is used to distinguish discrete-time signals from continuous-time signals?

  1. \(x(t)\) for discrete-time, \(x[n]\) for continuous-time
  2. \(x[n]\) for discrete-time where \(n\) is an integer, \(x(t)\) for continuous-time where \(t\) is real-valued
  3. Both use \(x(t)\) with different subscripts
  4. \(x_d(t)\) for discrete-time, \(x_c(t)\) for continuous-time
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The correct answer is B. Discrete-time signals are represented as \(x[n]\) where \(n\) is an integer-valued index, while continuous-time signals are represented as \(x(t)\) where \(t\) is a real-valued continuous variable. The square bracket notation \(x[n]\) emphasizes that the signal exists only at specific discrete time instances, typically at integer multiples of some sampling period.

Concept Tested: Discrete-Time Signals, Continuous-Time Signals

See: Continuous-Time vs. Discrete-Time Signals

3. What condition must a periodic signal satisfy?

  1. \(x(t) = -x(-t)\) for all \(t\)
  2. \(x(t) = x(-t)\) for all \(t\)
  3. \(x(t) = x(t + T)\) for all \(t\), where \(T\) is the fundamental period
  4. \(\int_{-\infty}^{\infty} |x(t)|^2 dt < \infty\)
Show Answer

The correct answer is C. Periodic signals must satisfy \(x(t) = x(t + T)\) for all values of \(t\), where \(T\) is the fundamental period. This means the signal repeats its values at regular intervals. The frequency \(f = 1/T\) describes how many complete cycles occur per unit time. Common examples include sinusoids, square waves, and carrier signals in radio transmission.

Concept Tested: Periodic Signals

See: Temporal Properties: Periodicity

4. What is the definition of the unit step function \(u(t)\) for \(t \geq 0\)?

  1. \(u(t) = 1\)
  2. \(u(t) = 0\)
  3. \(u(t) = t\)
  4. \(u(t) = \infty\)
Show Answer

The correct answer is A. The unit step function is defined as \(u(t) = 1\) for \(t \geq 0\) and \(u(t) = 0\) for \(t < 0\). This function represents an instantaneous transition from zero to one at time zero and serves as a building block for constructing more complex signals and modeling switching operations in circuits and control systems.

Concept Tested: Unit Step Function

See: Unit Step Function

5. How can any arbitrary signal be decomposed into even and odd components?

  1. Using Fourier transform only
  2. Using the formulas \(x_e(t) = \frac{x(t) + x(-t)}{2}\) and \(x_o(t) = \frac{x(t) - x(-t)}{2}\)
  3. By separating positive and negative amplitude values
  4. Using integration over the entire time domain
Show Answer

The correct answer is B. Any arbitrary signal can be decomposed into an even component using \(x_e(t) = \frac{x(t) + x(-t)}{2}\) and an odd component using \(x_o(t) = \frac{x(t) - x(-t)}{2}\). The original signal can be reconstructed as \(x(t) = x_e(t) + x_o(t)\). This decomposition is valuable in many analytical contexts, particularly for understanding signal behavior under various transformations.

Concept Tested: Even Signals, Odd Signals

See: Symmetry Properties

6. What is the key property of the unit impulse (Dirac delta) function \(\delta(t)\)?

  1. It has infinite duration and unit amplitude
  2. It equals 1 at \(t = 0\) and 0 everywhere else
  3. It satisfies the sifting property: \(\int_{-\infty}^{\infty} f(t)\delta(t-t_0) dt = f(t_0)\)
  4. It represents a rectangular pulse of unit area
Show Answer

The correct answer is C. The unit impulse function is defined through its sifting property: \(\int_{-\infty}^{\infty} f(t)\delta(t-t_0) dt = f(t_0)\). This represents an infinitely narrow, infinitely tall pulse with unit area. The impulse response of a system (its output when the input is an impulse) completely characterizes the system's behavior for all possible inputs through convolution operations.

Concept Tested: Unit Impulse Function

See: Unit Impulse Function

7. For a sinusoidal signal \(x(t) = A\cos(\omega t + \phi)\), what does the parameter \(\omega\) represent?

  1. The amplitude in volts
  2. The phase angle in radians
  3. The period in seconds
  4. The angular frequency in radians per second
Show Answer

The correct answer is D. In the sinusoidal signal \(x(t) = A\cos(\omega t + \phi)\), the parameter \(\omega\) represents the angular frequency in radians per second, where \(\omega = 2\pi f\) and \(f\) is the frequency in Hertz. The amplitude is represented by \(A\), the phase angle by \(\phi\), and the period is \(T = 2\pi/\omega = 1/f\).

Concept Tested: Sinusoidal Signals, Signal Frequency

See: Sinusoidal Signals

8. Given a signal \(x(t)\), what is the effect of the time shifting operation \(y(t) = x(t - 3)\)?

  1. The signal is compressed by a factor of 3
  2. The signal is delayed (shifted right) by 3 time units
  3. The signal is advanced (shifted left) by 3 time units
  4. The signal amplitude is multiplied by 3
Show Answer

The correct answer is B. The operation \(y(t) = x(t - t_0)\) with \(t_0 = 3 > 0\) delays the signal, shifting it to the right by 3 time units. A positive value of \(t_0\) always delays the signal, while a negative value would advance it (shift left). Time shifting translates a signal forward or backward in time without changing its shape.

Concept Tested: Time Shifting

See: Time Shifting

9. If you apply time scaling to signal \(x(t)\) to create \(y(t) = x(2t)\), what happens to the signal?

  1. The signal is expanded (plays slower) and frequencies decrease
  2. The signal is compressed (plays faster) and frequencies increase by factor 2
  3. The signal is reversed in time
  4. The signal amplitude is doubled
Show Answer

The correct answer is B. When \(y(t) = x(at)\) with \(a = 2 > 1\), the signal is compressed (plays faster), and all frequencies are increased by factor 2. Time scaling with \(a > 1\) compresses the signal, while \(0 < a < 1\) would expand it. The operation affects both the duration and frequency content of signals, with compression increasing frequencies and expansion decreasing them proportionally.

Concept Tested: Time Scaling

See: Time Scaling

10. What distinguishes energy signals from power signals?

  1. Energy signals have infinite energy and zero power; power signals have finite power
  2. Energy signals have finite total energy and zero average power; power signals have infinite energy but finite average power
  3. Energy signals are always periodic; power signals are always aperiodic
  4. Energy signals exist only in digital systems; power signals exist only in analog systems
Show Answer

The correct answer is B. Energy signals possess finite total energy (\(E = \int_{-\infty}^{\infty} |x(t)|^2 dt < \infty\)) and consequently have zero average power. Power signals have infinite energy but finite average power (\(P = \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} |x(t)|^2 dt < \infty\)). Energy signals are necessarily time-limited or decay rapidly, while periodic signals constitute the primary class of power signals due to their indefinite repetition.

Concept Tested: Energy Signals, Power Signals

See: Energy and Power Classifications