DFT, DTFT | N-Point Discrete Fourier Transform (DFT) 

Signals and Their Representations

• There are two main types of signals: continuous-time and discrete-time signals.

  • Continuous-time signals can be periodic or aperiodic.

    • Periodic signals repeat themselves after a fixed time period (T), and can be represented as a linear combination of sine and cosine functions using the Fourier series.

    • Aperiodic signals do not have a repeating pattern. They are treated as periodic signals with a time period tending towards infinity. Applying the Fourier series method to aperiodic signals results in the Fourier transform, representing the signal as a linear combination of sines and cosines.

  • Discrete-time signals have values at discrete points (e.g., at n=0, 1, 2, 3…).

    • These signals can be expressed as a linear combination of complex exponentials, which is called the Discrete-Time Fourier Transform (DTFT).

Discrete-Time Fourier Transform (DTFT)

• The DTFT transforms a discrete-time signal (x[n]) into a continuous frequency spectrum, X(e^jω).

  • The DTFT is defined by the following transform pair:

    • Analysis Equation (DTFT): X(e^jω) = Σ [from n = -∞ to ∞] x[n] * e^(-jωn)

    • Synthesis Equation (Inverse DTFT): x[n] = (1/2π) ∫ [over one cycle] X(e^jω) * e^(jωn) dω

    • Where:

      • x[n] is the discrete-time signal.

      • X(e^jω) is the DTFT, a continuous function of frequency (ω).

• The DTFT of a discrete-time signal results in a continuous frequency spectrum. This means that the frequency component (ω) is continuous.

• Example: For the discrete signal x[n] = a^n * u[n] (where |a| < 1), the DTFT is X(e^jω) = 1 / (1 - a*e^(-jω)). This transform has a magnitude and an angle that vary continuously with frequency.

The Need for Discrete Fourier Transform (DFT)

• Digital systems cannot process continuous frequency spectra, which is the result of a DTFT. Thus, a transformation into a discrete frequency spectrum is needed.

• To achieve a discrete frequency spectrum, the DTFT (X(e^jω)) is sampled at discrete frequency intervals.

• This sampling process leads to the Discrete Fourier Transform (DFT).

Discrete Fourier Transform (DFT)

• The DFT is used to transform a discrete-time signal into a discrete frequency spectrum.

• N-Point DFT: The DTFT is sampled at N equally spaced frequency intervals between 0 and 2π to obtain an N-point DFT.

  • This creates N discrete frequency points.

• The discrete frequencies (ωk) are given by: ωk = 2πk/N, where k = 0, 1, 2, ..., N-1.

• DFT Definition:

  • Analysis Equation (DFT): X[k] = Σ [from n = 0 to N-1] x[n] * e^(-j2πkn/N)

    • Where:

      • X[k] is the k-th frequency component of the DFT

      • x[n] is the discrete-time input signal.

      • N is the number of points in the DFT

  • Synthesis Equation (Inverse DFT): x[n] = (1/N) * Σ [from k = 0 to N-1] X[k] * e^(j2πkn/N)

• Conditions for N: If the length of the input sequence x[n] is L, then N must be greater than or equal to L (N ≥ L) to avoid aliasing.

Example of 4-Point DFT

• Consider a sequence x[n] = 1/3 for n = 0, 1, 2, and 0 otherwise. L = 3.

• To perform a 4-point DFT (N=4), use the following:

  • X[k] = Σ [from n = 0 to 3] x[n] * e^(-j2πkn/4), which simplifies to

  • X[k] = 1/3 * ( 1 + e^(-jπk/2) + e^(-jπk) )

• The DFT coefficients X[k] for k = 0, 1, 2, and 3 are computed as follows:

  • X = 1 (magnitude 1, angle 0)

  • X = 1/3 * e^(-jπ/2) = 1/3 * (cos(π/2) - j sin(π/2) ) (magnitude 1/3, angle -π/2)

  • X = 1/3 (magnitude 1/3, angle 0)

  • X = 1/3 * e^(jπ/2) = 1/3 * (cos(π/2) + j sin(π/2) ) (magnitude 1/3, angle π/2)

• The magnitude and phase of the DFT coefficients are then plotted against k.

• The magnitude spectrum shows the amplitude of each frequency component, while the phase spectrum shows the phase shift at each frequency.

Example of 8-Point DFT

• For an 8-point DFT (N=8) of the same sequence x[n]:

  • X[k] = Σ [from n = 0 to 7] x[n] * e^(-j2πkn/8)

  • X[k] = 1/3 * (1 + e^(-jπk/4) + e^(-jπk/2))

• You would then compute X[k] for k = 0, 1, 2, 3, 4, 5, 6, and 7 to get the 8 points.

10 numerical questions and answers focusing on key concepts of DFT, DTFT, and signal processing:

1. Question: What is the value of n when calculating a 4-point DFT?

   • Answer: For a 4-point DFT, the value of n ranges from 0 to 3.

2. Question: If a discrete-time signal x[n] has a length L of 5, what is the minimum value of N for an N-point DFT to avoid aliasing?

   • Answer: The minimum value of N must be greater than or equal to L. Therefore, the minimum N is 5.

3. Question: In the formula ωk = 2πk/N, what does k represent?

   • Answer: k represents the discrete frequency index, taking values from 0 to N-1.

4. Question: What is the value of ωk when k is 0 in a DFT?

   • Answer: When k is 0, ωk is 0, because ωk = 2π(0)/N = 0.

5. Question: If N is 8, what is the value of ωk when k is 2?

   • Answer: Using the formula ωk = 2πk/N, when k is 2 and N is 8, ωk = 2π(2)/8 = π/2.

6. Question: For a discrete signal x[n] = 1/3 for n = 0, 1, 2, what is the length L of the signal?

   • Answer: The length L of the signal is 3, as the signal has values for n = 0, 1, and 2.

7. Question: What is the value of X when computing a 4-point DFT for the sequence x[n] where *x[n] = 1/3 for n = 0, 1, 2?

   • Answer: X is calculated to be 1.

8. Question: For the 4-point DFT of x[n] = 1/3 for n = 0, 1, 2, what is the magnitude of X?

   • Answer: The magnitude of X is 1/3.

9. Question: For the 4-point DFT of x[n] = 1/3 for n = 0, 1, 2, what is the angle of X?

   • Answer: The angle of X is -π/2.

10. Question: In an 8-point DFT, how many discrete frequency points (values of k) are considered?

    • Answer: In an 8-point DFT, there are 8 discrete frequency points (k = 0 to 7).

5 long numerical questions and answers focusing on a deeper understanding of DFT calculations:

1. Question: A discrete-time signal x[n] is defined as x = 1, x = 2, and x = 1. Compute the 4-point DFT, X[k], and explicitly show the values of X, X, X, and X in both rectangular and polar forms, including the magnitude and phase.

   • Answer:

     • The 4-point DFT is given by:

     • X[k] = Σ [from n = 0 to 3] x[n] * e^(-j2πkn/4)

     • Since x[n] is only defined for n = 0, 1, and 2, we have x = 0.

     • X:

       • X = xe^(0) + xe^(0) + xe^(0) + xe^(0)

       • X = 1 + 2 + 1 + 0 = 4

       • In polar form: 4∠0

     • X:

       • X = xe^(0) + xe^(-jπ/2) + xe^(-jπ) + xe^(-j3π/2)

       • X = 1 + 2*(-j) + 1*(-1) + 0

       • X = 1 - 2j - 1 = -2j

       • In polar form: 2∠-π/2

     • X:

       • X = xe^(0) + xe^(-jπ) + xe^(-j2π) + xe^(-j3π)

       • X = 1 + 2*(-1) + 1*(1) + 0

       • X = 1 - 2 + 1 = 0

       • In polar form: 0∠0

     • X:

       • X = xe^(0) + xe^(-j3π/2) + xe^(-j3π) + xe^(-j9π/2)

       • X = 1 + 2*(j) + 1*(-1) + 0

       • X = 1 + 2j - 1= 2j

       • In polar form: 2∠π/2

2. Question: A discrete-time sequence x[n] is given by x[n] = {2, 1, -1, 0}. Determine the magnitude and phase spectra of its 4-point DFT, and show the magnitude and phase values for each frequency point (k=0, 1, 2, 3).

   • Answer:

     • We use the DFT formula:

     • X[k] = Σ [from n = 0 to 3] x[n] * e^(-j2πkn/4)

     • X:

       • X = 2e^(0) + 1e^(0) + (-1)e^(0) + 0e^(0) = 2 + 1 - 1 = 2

       • Magnitude: 2, Phase: 0

     • X:

       • X = 2e^(0) + 1e^(-jπ/2) + (-1)e^(-jπ) + 0e^(-j3π/2)

       • X = 2 + 1*(-j) + (-1)*(-1) = 2 - j + 1 = 3 - j

       • Magnitude: |3 - j| = √(3² + (-1)²) = √10, Phase: tan⁻¹(-1/3) = -0.322 radians

     • X:

       • X = 2e^(0) + 1e^(-jπ) + (-1)e^(-j2π) + 0e^(-j3π)

       • X = 2 + 1*(-1) + (-1)*(1) = 2 - 1 - 1 = 0

       • Magnitude: 0, Phase: undefined (or 0)

     • X:

       • X = 2e^(0) + 1e^(-j3π/2) + (-1)e^(-j3π) + 0e^(-j9π/2)

       • X = 2 + 1*(j) + (-1)*(-1) = 2 + j + 1 = 3 + j

       • Magnitude: |3 + j| = √(3² + 1²) = √10, Phase: tan⁻¹(1/3) = 0.322 radians

     • Magnitude Spectrum: |X| = 2, |X| = √10, |X| = 0, |X| = √10

     • Phase Spectrum: ∠X = 0, ∠X = -0.322 radians, ∠X is undefined, ∠X = 0.322 radians

3. Question: Consider a sequence x[n] = 1 for n=0,1,2,3 and x[n] = 0 otherwise. Calculate the 8-point DFT, X[k], and determine the value of X.

   • Answer:

     • The 8-point DFT is:

     • X[k] = Σ [from n = 0 to 7] x[n] * e^(-j2πkn/8)

     • Since x[n] is 1 for n=0 to 3 and 0 otherwise,

     • X[k] = Σ [from n = 0 to 3] 1 * e^(-j2πkn/8)

     • To calculate X:

     • X = Σ [from n = 0 to 3] e^(-j2π(2)n/8)

     • X = e^(0) + e^(-jπ/2) + e^(-jπ) + e^(-j3π/2)

     • X = 1 - j - 1 + j

     • X = 0

4. Question: If a discrete signal x[n] = {1, 2, 3, 4} find the 4-point DFT and then compute the inverse DFT to verify that you get back the original sequence.

   • Answer:

     • 4-point DFT:

       • X[k] = Σ [from n = 0 to 3] x[n] * e^(-j2πkn/4)

       • X = 1 + 2 + 3 + 4 = 10

       • X = 1 + 2*(-j) + 3*(-1) + 4*(j) = 1 - 2j - 3 + 4j = -2 + 2j

       • X = 1 + 2*(-1) + 3*(1) + 4*(-1) = 1 - 2 + 3 - 4 = -2

       • X = 1 + 2*(j) + 3*(-1) + 4*(-j) = 1 + 2j - 3 - 4j = -2 - 2j

       • So, X[k] = {10, -2 + 2j, -2, -2 - 2j}

     • Inverse 4-point DFT:

       • x[n] = (1/4) * Σ [from k = 0 to 3] X[k] * e^(j2πkn/4)

       • x:

         • x = (1/4) * (10 + (-2+2j) + (-2) + (-2-2j) ) = (1/4) * (10 - 2 + 2j - 2 - 2 -2j) = (1/4)*4 = 1

       • x:

         • x = (1/4) * (10 + (-2+2j)e^(jπ/2) + (-2)e^(jπ) + (-2-2j)e^(j3π/2) )

         • x = (1/4) * (10 + (-2+2j)j + (-2)(-1) + (-2-2j)(-j) )

         • x = (1/4) * (10 -2j - 2 + 2 + 2j - 2) = (1/4) * (10 - 2 - 2 - 2j +2j) = 8/4 = 2

       • x:

         • x = (1/4) * (10 + (-2+2j)e^(jπ) + (-2)e^(j2π) + (-2-2j)e^(j3π) )

         • x = (1/4) * (10 + (-2+2j)(-1) + (-2)(1) + (-2-2j)(-1) )

         • x = (1/4) * (10 + 2 - 2j - 2 + 2+2j) = (1/4) * 12/4= 3

       • x:

         • x = (1/4) * (10 + (-2+2j)e^(j3π/2) + (-2)e^(j3π) + (-2-2j)e^(j9π/2) )

         • x = (1/4) * (10 + (-2+2j)(-j) + (-2)(-1) + (-2-2j)(j) )

         • x = (1/4) * (10 + 2j + 2 - 2+ 2j) = (1/4) * (16/4) = 4

       • The inverse DFT gives x[n] = {1, 2, 3, 4}, verifying the original sequence.

5. Question: A continuous signal is sampled to produce a discrete sequence x[n] = cos(πn/4) for n = 0 to 7. Compute the 8-point DFT X[k], and find the value of X.

   • Answer:

     • The 8-point DFT is:

     • X[k] = Σ [from n = 0 to 7] x[n] * e^(-j2πkn/8)

     • x[n] = cos(πn/4), so

     • X[k] = Σ [from n = 0 to 7] cos(πn/4) * e^(-j2πkn/8)

     • We want to find X, so k = 1:

     • X = Σ [from n = 0 to 7] cos(πn/4) * e^(-j2πn/8)

     • X = Σ [from n = 0 to 7] cos(πn/4) * e^(-jπn/4)

     • X = cos(0) * e^0 + cos(π/4) * e^(-jπ/4) + cos(π/2) * e^(-jπ/2) + cos(3π/4) * e^(-j3π/4) + cos(π) * e^(-jπ) + cos(5π/4) * e^(-j5π/4) + cos(3π/2) * e^(-j3π/2) + cos(7π/4) * e^(-j7π/4)

     • X = 1 + (1/√2) * (1/√2 - j/√2) + 0 + (-1/√2) * (-1/√2 - j/√2) + (-1) * (-1) + (-1/√2) * (-1/√2 + j/√2) + 0 + (1/√2) * (1/√2 + j/√2)

     • X = 1 + (1/2 - j/2) + 0 + (1/2 + j/2) + 1 + (1/2 - j/2) + 0 + (1/2 + j/2)

     • X = 1 + 1/2 - j/2 + 1/2 + j/2 + 1 + 1/2 - j/2 + 1/2 + j/2

     • X = 1 + 1 + 1/2 + 1/2 +1/2 + 1/2

     • X = 1 + 1 + 1 + 1 = 4