Discrete-Time Fourier Analysis Lecture 3

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.