Properties of Discrete Fourier Transform (DFT)
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I. DFT Basics
The Discrete Fourier Transform (DFT) is a mathematical tool that decomposes a discrete-time sequence into its constituent frequencies. It is a fundamental concept in digital signal processing.
A discrete-time sequence x[n] is transformed into its DFT representation X[k], where k represents the frequency index. The DFT pair is denoted as x[n] ↔ X[k].
For an N-point DFT, n ranges from 0 to N-1 and k ranges from 0 to N-1.
The length of the input sequence x[n] is L. The number of points, N, for the DFT must be greater than or equal to L (N ≥ L). This ensures that no information is lost and avoids aliasing.
The basic DFT formula is: X[k] = Σ [from n = 0 to N-1] x[n] * e^(-j2πkn/N), where:
X[k] is the DFT of the sequence.
x[n] is the input discrete-time sequence.
N is the length of the DFT.
k is the frequency index.
n is the time index.
The inverse DFT formula (IDFT) to get back the original sequence is: x[n] = (1/N) * Σ [from k = 0 to N-1] X[k] * e^(j2πkn/N) [from our conversation history].
II. DFT Properties
The source discusses several key properties of the DFT, which are useful in signal processing:
Linearity: The DFT of a linear combination of sequences is the same linear combination of their respective DFTs. If a and b are constants, then: DFT{ax₁[n] + bx₂[n]} = aX₁[k] + bX₂[k].
Periodicity: Both the discrete-time signal x[n] and its DFT X[k] are periodic with period N. This means that x[n+N] = x[n] and X[k+N] = X[k].
Circular Time Shift: If x[n] is shifted circularly by m units to become x[n-m], its DFT becomes X[k] multiplied by e^(-j2πkm/N).
DFT{x[n-m]} = X[k] e^(-j2πkm/N)
Time Reversal: If the sequence x[n] is time-reversed to x[-n], its DFT also reverses such that DFT{x[-n]} = X[-k]. For an N-point DFT, time reversal is equivalent to x[N-n] and DFT{x[N-n]} = X[N-k].
Conjugation: The DFT of the conjugate of a sequence x[n]** is not just the conjugate of its DFT. It is given by: DFT{x[n]** } = X[N-k]**. Thus, the DFT of a conjugated sequence is time-reversed and then conjugated.
Circular Frequency Shift: If the sequence x[n] is multiplied by e^(j2πmn/N), its DFT is shifted by m units in the frequency domain. So, DFT{x[n] e^(j2πmn/N)} = X[k-m].
Multiplication of Sequences: If two sequences x₁[n] and x₂[n] are multiplied, the DFT of the resulting sequence is a scaled circular convolution of their respective DFTs: DFT{x₁[n] x₂[n]} = (1/N) *X₁[k] ⊛ *X₂[k], where ⊛ represents circular convolution.
III. DFT Calculation Steps
The source and our conversation history show how to compute a DFT.
Given a sequence x[n], determine N, the length of the DFT. N must be at least the length of x[n].
Use the DFT formula: X[k] = Σ [from n = 0 to N-1] x[n] * e^(-j2πkn/N) to calculate X[k] for k = 0, 1, 2, ...., N-1.
This involves summing the products of each sample of x[n] with a complex exponential term.
The complex exponential e^(-j2πkn/N) can be expanded into cosine and sine terms using Euler’s formula: e^(-jθ) = cos(θ) - j sin(θ).
Calculate DFT values X, X, X, ...., X[N-1] by putting in the k values.
Express the DFT values in rectangular form (a + jb) [from our conversation history].
Convert to polar form (magnitude and phase):
Magnitude = √(a² + b²) [from our conversation history]
Phase = tan⁻¹(b/a) [from our conversation history]
The magnitude spectrum is |*X[k]*| for k = 0, 1, 2,...., N-1
The phase spectrum is ∠X[k] for k = 0, 1, 2,....., N-1
Plot the magnitude and phase spectra to visualize the frequency content of the signal.
IV. Examples from the Conversation History
The conversation history includes several numerical examples:
Calculating a 4-point DFT for a given sequence [from our conversation history]
Computing the magnitude and phase spectra for specific sequences [from our conversation history]
Verifying the inverse DFT to retrieve the original sequence [from our conversation history]
These examples demonstrate the step-by-step process of calculating DFT values and interpreting the magnitude and phase spectra [from our conversation history].
V. Key Points to Remember
The DFT is a fundamental tool for analyzing the frequency content of discrete-time signals.
Understanding the properties of the DFT allows for efficient manipulation and processing of signals.
DFT calculations involve complex exponentials, which can be broken down into sine and cosine terms using Euler’s formula.
The magnitude spectrum shows the strength of different frequency components in the signal, while the phase spectrum shows the phase shift of these components.
Short Question-Answers:
Q1: What is the Discrete Fourier Transform (DFT) used for?
A: The DFT is a mathematical tool used to decompose a discrete-time sequence into its constituent frequencies. It's a fundamental concept in digital signal processing [from our conversation history].
Q2: What is the relationship between the length of the input sequence and the number of points in a DFT?
A: If the input sequence x[n] has a length of L, the number of points N in the DFT should be greater than or equal to L (N ≥ L). This ensures that no information is lost during the transformation and avoids aliasing [from our conversation history].
Q3: What does the linearity property of DFT mean?
A: The linearity property states that the DFT of a linear combination of sequences is the same linear combination of their individual DFTs. So, DFT{ax₁[n] + bx₂[n]} = aX₁[k] + bX₂[k], where a and b are constants [1, from our conversation history].
Q4: What does it mean that the DFT is periodic?
A: Both the discrete-time signal x[n] and its DFT X[k] are periodic with a period of N, where N is the number of points in the DFT. This means that x[n+N] = x[n] and X[k+N] = X[k] [1, from our conversation history].
Q5: How does a circular time shift in the sequence affect its DFT?
A: If a sequence x[n] is circularly shifted by m units to become x[n-m], its DFT becomes X[k] multiplied by e^(-j2πkm/N). This is known as the circular time shift property [2, from our conversation history].
Q6: What happens to the DFT when you reverse the time of the original sequence?
A: If you reverse a sequence x[n] to x[-n], the DFT is also reversed such that DFT{x[-n]} = X[-k]. For an N-point DFT, this is equivalent to DFT{x[N-n]} = X[N-k] [2, from our conversation history].
Q7: How does the DFT of a conjugated sequence differ from the conjugate of the DFT?
A: The DFT of the conjugate of a sequence x[n] is not just the conjugate of its DFT, X[k]. Instead, DFT{x[n] } = X[N-k]**. The DFT is conjugated and time-reversed [2, from our conversation history].
Q8: How do you calculate the DFT of a sequence in practice?
A: To calculate the DFT of a sequence, you use the DFT formula: X[k] = Σ [from n = 0 to N-1] x[n] * e^(-j2πkn/N). You evaluate this sum for each value of k from 0 to N-1. The complex exponential can be expanded into cosine and sine terms using Euler's formula [3, from our conversation history].
Q9: What are magnitude and phase spectra, and what information do they provide?
A: The *magnitude spectrum, |*X[k]|, shows the strength of different frequency components in the signal. The phase spectrum, ∠X[k], shows the phase shift of these components. Together, they form the complete frequency spectrum of the signal [4, from our conversation history].
Q10: If you multiply two sequences in the time domain, what is the equivalent operation in the frequency domain?
A: If two sequences x₁[n] and x₂[n] are multiplied, the DFT of the result is a scaled circular convolution of their individual DFTs. This is given by: DFT{x₁[n] x₂[n]} = (1/N) *X₁[k] ⊛ *X₂[k], where ⊛ represents circular convolution [3, from our conversation history].
Numerical Question-Answers:
Q1: A sequence x[n] = {1, 2, 3, 4} is given. Calculate its 4-point DFT, X[k], and show the magnitude and phase spectrum.
A:
We use the DFT formula: X[k] = Σ [from n = 0 to N-1] x[n] * e^(-j2πkn/N), where N=4
X: X = 1e⁰ + 2e⁰ + 3e⁰ + 4e⁰ = 1 + 2 + 3 + 4 = 10
X: X = 1e⁰ + 2e^(-jπ/2) + 3e^(-jπ) + 4e^(-j3π/2) = 1 - 2j - 3 + 4j = -2 + 2j
X: X = 1e⁰ + 2e^(-jπ) + 3e^(-j2π) + 4e^(-j3π) = 1 - 2 + 3 - 4 = -2
X: X = 1e⁰ + 2e^(-j3π/2) + 3e^(-j3π) + 4e^(-j9π/2) = 1 + 2j - 3 - 4j = -2 - 2j
Magnitude Spectrum:
|X| = |10| = 10
|X| = |-2 + 2j| = √( (-2)² + 2²) = √8 = 2√2
|X| = |-2| = 2
|X| = |-2 - 2j| = √((-2)² + (-2)²) = √8 = 2√2
Phase Spectrum:
∠X = 0
∠X = tan⁻¹(2/-2) = 3π/4
∠X = π
∠X = tan⁻¹(-2/-2) = -3π/4
Q2: A sequence x[n] = {1, -1, 1, -1} is given. Calculate its 4-point DFT X[k].
A:
Using the DFT formula with N=4, we have:
X: 1*e⁰ + (-1)e⁰ + 1e⁰ + (-1)*e⁰ = 1 - 1 + 1 - 1 = 0
X: 1*e⁰ + (-1)e^(-jπ/2) + 1e^(-jπ) + (-1)*e^(-j3π/2) = 1 + j - 1 - j = 0
X: 1*e⁰ + (-1)e^(-jπ) + 1e^(-j2π) + (-1)*e^(-j3π) = 1 + 1 + 1 + 1 = 4
X: 1*e⁰ + (-1)e^(-j3π/2) + 1e^(-j3π) + (-1)*e^(-j9π/2) = 1 - j - 1 + j = 0
Therefore, the DFT X[k] = {0, 0, 4, 0}.
Q3: A sequence x[n] has a DFT X[k] = {1, 2, 1, 0}. If a new sequence y[n] is created by a circular time shift of x[n] by 1 unit (y[n] = x[n-1]), find the DFT of y[n], Y[k].
A:
We use the circular time shift property: DFT{x[n-m]} = X[k] * e^(-j2πkm/N). Here, m = 1 and N = 4.
Y[k] = X[k] * e^(-j2πk/4) = X[k] * e^(-jπk/2)
Y = X * e⁰ = 1 * 1 = 1
Y = X * e^(-jπ/2) = 2 * (-j) = -2j
Y = X * e^(-jπ) = 1 * (-1) = -1
Y = X * e^(-j3π/2) = 0 * j = 0
Therefore, Y[k] = {1, -2j, -1, 0}
Q4: If x[n] has a DFT X[k] = {4, 0, 0, 0}, what is the DFT of the time-reversed sequence x[-n]?
A:
We use the time reversal property: DFT{x[-n]} = X[-k] which for N=4 is equivalent to X[N-k].
X[-0] = X = 4
X[-1] = X = 0
X[-2] = X = 0
X[-3] = X = 0
Therefore the DFT of x[-n] is {4, 0, 0, 0}.
Q5: A sequence x[n] has a DFT X[k]. If y[n] = x[n] * cos(πn/2), find the DFT of y[n].
A:
We express cos(πn/2) using Euler's formula: cos(πn/2) = (1/2) * [e^(jπn/2) + e^(-jπn/2)]
So, y[n] = x[n] * (1/2) * [e^(jπn/2) + e^(-jπn/2)]. This is a linear combination of frequency shifts
y[n] = (1/2) * x[n] * e^(jπn/2) + (1/2) * x[n] * e^(-jπn/2)
Using the frequency shift property: DFT{x[n] * e^(j2πmn/N)} = X[k-m]. Here, m = 1 and N=4
Therefore, Y[k] = (1/2) * X[k-1] + (1/2) * X[k+1], where X[-1] is interpreted as X and X as X.
The DFT of y[n] will be a combination of shifted versions of X[k], scaled by 1/2. If we are given X[k], we can compute Y[k].
Q6: Given x[n] = {1, 2, 1, 2} and h[n] = {1, 0, 1, 0}, find the DFT of the sequence y[n] = x[n] * h[n] (element-wise multiplication).
A:
First, calculate X[k] and H[k]:
X = 6; X = 0; X = -2; X = 0; X[k] = {6, 0, -2, 0}
H = 2; H = 0; H = 2; H = 0; H[k] = {2, 0, 2, 0}
Use the multiplication property of DFT: DFT{x[n] * h[n]} = (1/N) * X[k] ⊛ H[k] where ⊛ is circular convolution.
Calculate the circular convolution:
Y = (1/4) * [ (62) + (00) + (-22) + (00) ] = 1/4 * [12 + 0 -4 + 0] = 2
Y = (1/4) * [ (60) + (02) + (-20) + (02) ] = 0
Y = (1/4) * [ (62) + (00) + (-22) + (00) ] = 2
Y = (1/4) * [ (60) + (02) + (-20) + (02) ] = 0
Therefore, DFT of y[n] = {2, 0, 2, 0}
Q7: A sequence x[n] = {1, 2, 3} is given. Calculate its 4-point DFT by padding with a zero and also show the magnitude and phase spectrum.
A:
We pad x[n] with a zero to make it a 4-point sequence: x[n] = {1, 2, 3, 0}
We use the DFT formula: X[k] = Σ [from n = 0 to N-1] x[n] * e^(-j2πkn/N), where N = 4
X: X = 1 + 2 + 3 + 0 = 6
X: X = 1 + 2e^(-jπ/2) + 3e^(-jπ) = 1 - 2j - 3 = -2 - 2j
X: X = 1 + 2e^(-jπ) + 3e^(-j2π) = 1 - 2 + 3 = 2
X: X = 1 + 2e^(-j3π/2) + 3e^(-j3π) = 1 + 2j - 3 = -2 + 2j
Magnitude Spectrum:
|X| = 6
|X| = √((-2)² + (-2)²) = √8 = 2√2
|X| = 2
|X| = √((-2)² + 2²) = √8 = 2√2
Phase Spectrum:
∠X = 0
∠X = tan⁻¹(-2/-2) = -3π/4
∠X = 0
∠X = tan⁻¹(2/-2) = 3π/4
Q8: A sequence x[n] has a DFT X[k] = {1, 0, 1, 0}. If y[n] = x[n]*(-1)^n, find Y[k].
A:
Note that (-1)^n is equal to e^(jπn). So y[n] = x[n] * e^(jπn)
We use the circular frequency shift property with m=2 and N =4: DFT{x[n] * e^(j2πmn/N)} = X[k-m]
Therefore, Y[k] = X[k-2], which represents a circular shift in the frequency domain.
Y = X = X[-2] = X = 1
Y = X = X[-1] = X = 0
Y = X = X = 1
Y = X = X = 0
The DFT of y[n], Y[k] = {1, 0, 1, 0}
*Q9: If X[k] is the DFT of x[n], what is the DFT of x[n]cos(2πn/4), given N=4?
A:
We have x[n] multiplied by cos(2πn/4), which is cos(πn/2).
Using Euler's identity, cos(πn/2) = 1/2 * [e^(jπn/2) + e^(-jπn/2)].
So the sequence is x[n] * 1/2 * [e^(jπn/2) + e^(-jπn/2)].
This is a linear combination of frequency shifts.
We use the frequency shift property: DFT{x[n] * e^(j2πmn/N)} = X[k-m]. Here, N = 4, m= 1 (for e^(jπn/2)), m=-1 (for e^(-jπn/2))
Therefore, DFT{x[n]*cos(πn/2)} = 1/2 * [X[k-1] + X[k+1] ] .
If we are given the DFT X[k], we can calculate the shifted DFT Y[k].
Q10: Given a sequence x[n] = {1, 1, 1, 1}. Calculate its 4-point DFT, X[k]. Then, find the DFT of y[n] where y[n] = x[n] * x[n]. A:
First, find the DFT of x[n]:
X = 1 + 1 + 1 + 1 = 4
X = 1 - j - 1 + j = 0
X = 1 - 1 + 1 - 1 = 0
X = 1 + j - 1 - j = 0
X[k] = {4, 0, 0, 0}
Now find the DFT of y[n] = x[n] * x[n], which is simply {1, 1, 1, 1} in this case.
Since y[n] is the same as x[n], then Y[k] will also be same as X[k], so Y[k] = {4, 0, 0, 0}.
Alternatively, from the multiplication property DFT{x[n]*x[n]} = 1/N (X[k] ⊛ X[k]), 1/4 * circular convolution of X[k] with itself gives {4,0,0,0}.
These examples demonstrate how to apply the DFT and its properties to solve numerical problems.
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