Discrete Time

Question 1

List the signal types covered in the course so far.

Answer 1

Continuous-time vs. Discrete-time, Analog vs. Digital, Multidimensional vs. Unidimensional, Causal vs. Noncausal.

Question 2

List the signal properties covered in the course so far.

Answer 2

Periodic vs. Aperiodic, Even vs. Odd, Power vs. Energy Signal.

Question 3

What signal transformations and properties were studied?

Answer 3

Time-scaled, time-shifting, etc.; Sampling Property.

Question 4

What system properties were covered related to linearity and invariance?

Answer 4

Linear and time-invariant systems; Impulse and step responses; Convolution to find the output using the impulse response (analytical, properties, graphical).

Question 5

List additional system properties studied.

Answer 5

Causal/Non-causal, Dynamic/Static, Stable/Unstable; Impulse response of systems described by a Linear, Constant-Coefficient, Differential Equation (LCDE).

Question 6

What frequency analysis topics were covered?

Answer 6

Fourier Analysis: Fourier Series and Fourier Transform; Parseval’s relations; Properties; Applications like AM Radio Communication and Filters (Ideal and Practical).

Question 7

Define a continuous-time signal.

Answer 7

A signal defined for every instance in time, where time can take any real number; usually denoted as x(t). Example: Temperature of a room or body.

Question 8

Define a discrete-time signal.

Answer 8

A signal defined only for certain moments in time; usually denoted as x[n]. Example: Daily temperature (values provided once per day); Naturally discrete signals like step count, demographics, or total population.

Question 9

Why is sampling of continuous-time signals important for discrete-time signals?

Answer 9

It creates successive samples of an underlying continuous phenomenon; Modern digital processors require discrete-time sequences for systems like digital autopilots to digital audio systems.

Question 10

How is a discrete-time signal represented in terms of sampling?

Answer 10

x[0], x[1], x[2], … are values at integer multiples of a basic interval T_s, i.e., x(0 · T_s), x(1 · T_s), etc.; For non-integer n, the signal is undefined; DT signals are lists of numbers indexed by integers (vectors).

Question 11

Why is sampling theory important?

Answer 11

To facilitate signal analysis via computers, including frequency content analysis and digital filtering (removal of noise or frequency components).

Question 12

Describe the framework for sampling in signal processing.

Answer 12

Physical signal → Transducer → Signal conditioning → Analog-to-Digital Conversion (ADC) → Digital processing → Digital-to-Analog Conversion (DAC) → Transducer → Physical signal; Ensures time information in the analog signal is not lost, allowing recovery from samples.

Question 13

What is Shannon’s Sampling Theorem?

Answer 13

A bandlimited continuous-time signal x(t) with max frequency B Hz can be exactly recovered from samples x[n] = x(n · T_s) if F_s = 1/T_s > 2B (sampling frequency > twice the max frequency).

Question 14

What is another name for Shannon’s Sampling Theorem and its key implication?

Answer 14

Nyquist-Shannon sampling theorem or Nyquist theorem; Ensures no information is lost during digitization; Sampling rate F_s must be > 2 × B (max frequency in signal) for accurate reconstruction.

Question 15

How is sampling mathematically viewed?

Answer 15

As the product of continuous-time signal x(t) with a train of impulses: x_s(t) = x(t) · ∑_{n=-∞}^∞ δ(t - n T_s); Band-limited signal with max frequency B.

Question 16

What is the Fourier transform of the sampled signal x_s(t)?

Answer 16

X_s(f) = X(f) * S(f), where S(f) is the Fourier transform of the impulse train s(t) = ∑_{n=-∞}^∞ δ(t - n T_s).

Question 17

What is the Fourier transform of the impulse train s(t)?

Answer 17

S(f) = F_s ∑{n=-∞}^∞ δ(f - n F_s) or S(ω) = (2π / T_s) ∑{n=-∞}^∞ δ(ω - n (2π / T_s)); A train of impulses in the frequency domain.

Question 18

Express the spectrum of the sampled signal X_s(f).

Answer 18

X_s(f) = F_s ∑_{n=-∞}^∞ X(f - n F_s); Copies of the original spectrum X(f) spaced F_s apart, scaled by F_s.

Question 19

How can the original signal x(t) be reconstructed from samples?

Answer 19

Recover X(f) from X_s(f) by low-pass filtering if no overlap between cycles; Important for digital-to-analog converters.

Question 20

What condition prevents overlap in the frequency spectrum of the sampled signal?

Answer 20

F_s - B > B, so F_s > 2B; This completes the proof of the sampling theorem.

Question 21

What is the Nyquist frequency and Nyquist interval?

Answer 21

Nyquist frequency: F_s = 2B; Nyquist interval: T = 1/(2B) for a signal with max frequency B.

Question 22

What happens if the sampling frequency is not greater than 2B?

Answer 22

Overlap (aliasing) occurs in the frequency spectrum, preventing accurate recovery of the original signal.

Question 23

Give an example of a practical application of discrete-time signals from sampling.

Answer 23

Digital audio systems or digital autopilots, where continuous signals are sampled for processing.