Discrete-Time Fourier Analysis Lecture 3
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Questions and Answers

What is the relationship between the Fourier transform of a signal and its period?

  • The Fourier transform does not exhibit periodicity.
  • The Fourier transform is periodic in frequency. (correct)
  • The Fourier transform only applies to periodic signals.
  • The Fourier transform is periodic in time.

    • Which condition is NOT necessary for an LTI system's frequency response to be evaluated?

  • The input sequence must be periodic. (correct)
  • The sequence must be absolutely summable.
  • The system must be linear.
  • The system must be time-invariant.

    • What does the term A in the sinusoidal sequence x(n) represent?

  • Amplitude of the signal. (correct)
  • Summation limits.
  • Frequency of the series.
  • Phase shift of the signal.

    • In the context of the impulse response function h(n), what is its role in an LTI system?

    <p>It represents the system's behavior to a specific input signal.</p> Signup and view all the answers

    In Matlab implementation of an LTI system, which of the following represents the numerator of the transfer function?

    <p>b = [b0, b1, ..., bM]</p> Signup and view all the answers

    Which of the following best describes the property of symmetry in discrete-time Fourier transforms (DTFT)?

    <p>The DTFT of a real-valued signal is conjugate symmetric.</p> Signup and view all the answers

    What happens to a continuous-time signal when it is digitally sampled?

    <p>It may introduce aliasing if not sampled correctly.</p> Signup and view all the answers

    Which of the following statements about the frequency response function is false?

    <p>It can be computed without the impulse response.</p> Signup and view all the answers

    What is the period of the Discrete-Time Fourier Transform (DTFT) in the frequency domain?

    <p>$2 ext{π}$</p> Signup and view all the answers

    For a real-valued signal x(n), which property holds true about the DTFT?

    <p>X(e^{-jw}) = X^*(e^{jw})</p> Signup and view all the answers

    What is a necessary condition for the Discrete-Time Fourier Transform (DTFT) to exist?

    <p>The signal must be absolutely summable.</p> Signup and view all the answers

    When evaluating DTFT in MATLAB for a finite duration signal, what operation can be used?

    <p>Matrix-vector multiplication.</p> Signup and view all the answers

    In MATLAB, how can the DTFT be evaluated if the signal x(n) is of infinite duration?

    <p>Evaluate X over [0, π] frequencies.</p> Signup and view all the answers

    What are the complex values in the DTFT primarily used to represent?

    <p>Frequency magnitudes and angles</p> Signup and view all the answers

    What is a recommended practice when plotting the frequency in MATLAB for DTFT?

    <p>Plot in units of π</p> Signup and view all the answers

    Which of the following implies symmetry in the DTFT for a real-valued signal?

    <p>X(e^{-jw}) = X^*(e^{jw})</p> Signup and view all the answers

    What is the primary purpose of the sampling principle for band-limited signals?

    <p>To allow reconstruction of the signal from its samples</p> Signup and view all the answers

    If a signal has a bandwidth of $F_0$, what is the minimum required sampling frequency to avoid aliasing?

    <p>$F_s = 2F_0$</p> Signup and view all the answers

    What condition leads to aliasing in sampled signals?

    <p>Sampling frequency is less than twice the bandwidth</p> Signup and view all the answers

    In the context of sampling, what does the term 'band-limited' imply about a signal?

    <p>The signal has a finite frequency range with a specified cutoff</p> Signup and view all the answers

    Which mathematical relation is used to connect digital frequencies to analog frequencies in sampling?

    <p>$ ext{w} = ext{Ω}T_s$</p> Signup and view all the answers

    What is the expected outcome when Ts is much smaller than the inverse of the signal bandwidth?

    <p>The original signal can be accurately reconstructed</p> Signup and view all the answers

    Which effect does sampling have on the frequency response of a band-limited signal?

    <p>It creates an overlapping replica of the signal</p> Signup and view all the answers

    What is a limitation of ideal interpolation in signal reconstruction?

    <p>It is non-causal and not realizable</p> Signup and view all the answers

    Study Notes

    Discrete-Time Fourier Analysis (Lecture 3)

    • Discrete-time Fourier Analysis is covered in chapter 3.
    • Output of a linear system (y(n)) to input (x(n)) is given by a summation of the input multiplied by impulse response (L[8(n-k)]). The impulse response (h(n)) of an LTI system is the system's response to a unit impulse.
    • LTI system: Linear Time Invariant
    • The entire system can be represented in the terms of its response to a unit sample sequence (impulse response).
    • Any signal can be represented by a linear combination of scaled and delayed unit samples.
    • Discrete signals can be represented as a linear combination of basis signals. Different basis sets offer different advantages/disadvantages depending on the system being considered.
    • Discrete-Time Fourier Transform (DTFT) transforms a discrete signal (x[n]) into a complex-valued continuous function (X). Digital frequency (w) is measured in radians.
    • If x[n] is absolutely summable, then its DTFT exists, expressed as: X(ejω) = Σx(n)e-jωn n=-∞
    • Inverse DTFT: x(n) = (1/2π)∫⁻π⁺π X(ejω)ejωn dω
    • DTFT is periodic in ω with period 2π.
    • For real-valued x(n), X(ejω) is conjugate symmetric (X(e−jω) = X*(e^jω)), requiring only half the domain [0, π] to plot.

    Properties of DTFT

    • Linearity: DTFT is a linear transformation: F[ax₁(n) + bx₂(n)] = aF[x₁(n)] + bF[x₂(n)].
    • Time Shifting: Shifting a signal in the time domain corresponds to a phase shift in the frequency domain: F[x(n − k)] = X(ejω)e−jωk
    • Frequency Shifting: Multiplying a signal by a complex exponential in the time domain corresponds to a shift in the frequency domain: F[x(n)e^jω₀n] = X(e^(j(ω−ω₀)))
    • Conjugation: Conjugating a signal in the time domain corresponds to conjugating and folding in the frequency domain: F[x*(n)] = X*(e^−jω)
    • Folding: Folding a signal in the time domain corresponds to folding in the frequency domain: F[x(−n)] = X(e^−jω)
    • Symmetry (Real Sequences): For real-valued x(n), the real part of X(ejω) is even in ω, and the imaginary part of X(ejω) is odd in ω.
    • Convolution: F[x₁(n) * x₂(n)] = F[x₁(n)] * F[x₂(n)] = X₁(e^jω)X₂(e^jω)

    MATLAB Implementation

    • MATLAB can be used to evaluate and plot DTFT magnitude and phase/angle in frequency domain at equi-spaced points in [0, π].
    • MATLAB code snippets for plotting the various parts are provided in the original notes

    Frequency Response of LTI Systems

    • The Fourier transform of an LTI system's impulse response is the frequency response.
    • The frequency response determines how the system modifies the input signal at different frequencies.
    • The frequency response allows us to represent the output as a modification of the input by the system response.

    Sampling and Reconstruction

    • Analog signals can be converted to discrete-time using analog-to-digital conversion (ADC).
    • Discrete-time signals can be converted back to analog using digital-to-analog conversion (DAC).
    • The sampling process creates replicas of the continuous-time signal's frequency spectrum in the discrete-time signal's frequency spectrum.
    • The digital frequency (ω) is related to the analog frequency (Ω) by the sampling frequency (Fs) as w = ΩTs
    • Proper sampling (Fs > 2Fo) avoids overlap of replicas (aliasing).
    • Reconstruction uses an ideal low-pass filter to recover the original signal by removing the replicas.
    • Practical D/A conversion involves interpolation, such as zero-order hold, first-order hold, or cubic spline approximation. Each approach has differing accuracy.

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    Description

    This quiz covers key concepts from Lecture 3 on Discrete-Time Fourier Analysis. It focuses on the relationship between input and output of linear systems, explores impulse responses of LTI systems, and discusses the Discrete-Time Fourier Transform (DTFT). Test your understanding of these essential topics in signal processing.

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