Z-Transform and DFT: LTI System Analysis & Convolution 

Relationship between Z-transform and DFT

• The Z-transform of a sequence x[n] of length L is defined as X(z) = Σ x[n]z^(-n), where the sum is taken from n=0 to L-1.

• The expression e^(j2πk/N) represents points on the unit circle in the Z-plane. The magnitude of this expression is 1, and the angle is 2πk/N, where N is the number of samples and k ranges from 0 to N-1.

• When the Z-transform is evaluated at z = e^(j2πk/N), it becomes the Discrete Fourier Transform (DFT). This is because X(e^(j2πk/N)) = Σ x[n]e^(-j2πkn/N), which is the definition of the DFT.

  • Therefore, the DFT can be seen as a specific case of the Z-transform evaluated along the unit circle in the Z-plane.

Analysis of LTI Systems Using DFT

• An LTI system is characterized by its impulse response, h[n]. The output y[n] of an LTI system for an input x[n] is given by the convolution of the input with the impulse response: y[n] = x[n] * h[n], where * denotes convolution.

• In the context of DFT, a distinction must be made between linear convolution and circular convolution.

• Linear convolution is the standard convolution operation defined as y[n] = Σ x[m]h[n-m], where the sum is taken from m = -∞ to ∞.

• Circular convolution is the convolution operation that results when using DFT.

• To use DFT to analyze an LTI system, linear convolution must be converted to circular convolution. This can be achieved by using a specific technique that involves zero-padding the input and impulse response sequences before performing the DFT.

  • If x[n] has N1 samples and h[n] has N2 samples, the sequences should be padded with zeros to have a length of N1 + N2 - 1.

  • The DFTs of the padded sequences, X[k] and H[k], are computed.

  • The DFT of the output is then obtained by multiplying the individual DFTs: Y[k] = X[k] * H[k].

  • Finally, the inverse DFT of Y[k] gives the output sequence y[n], which corresponds to the linear convolution of x[n] and h[n].

In this video, we explore the relation between Z-Transform and Discrete Fourier Transform (DFT); how to analyze Linear Time-Invariant (LTI) systems using DFT, and the method to convert linear convolution into circular convolution with N-point DFT. These concepts are critical for students of electrical, electronics, communications, and computer science engineering, especially those focusing on digital signal processing (DSP) and signals and systems. Learn how these mathematical tools are applied to real-world engineering problems, including signal filtering and system analysis. 

Example of Circular and Linear Convolution The source provides an example of how to perform circular and linear convolution.

• Two sequences x[n] = {1, 2} and h[n] = {2, 1} are used.

• To find the circular convolution of these two sequences, we compute the DFT of each sequence and multiply them, then take the inverse DFT. With N=2,

  • X=3, X=-1

  • H=3, H=1

  • Y = XH = 9, Y = XH = -1

  • The resulting circular convolution is y[n] = {4, 5}.

• To find the linear convolution of these same two sequences, we first need to pad the sequences with zeros. We must find the number of samples that are equal to N1+N2-1, which is 2+2-1=3, so the sequences are extended with zeros to have a length of 3. Thus x[n] becomes {1, 2, 0} and h[n] becomes {2, 1, 0}.

• The DFT of each padded sequence is computed:

  • X=3, X=-0.5+j0.866, X=-0.5-j0.866

  • H=3, H=0.5+j0.866, H=0.5-j0.866

• The product of the two DFTs Y[k]=X[k]*H[k] is then computed.

  • Y = XH=9, Y = XH= -2, Y= X*H=-2.

• The inverse DFT is then computed, to give the linear convolution, y[n] = {2, 5, 2}.

In summary, the Z-transform provides a general framework, and the DFT is a special case when the Z-transform is evaluated on the unit circle. The DFT can be used to analyze LTI systems by converting linear convolution into circular convolution using zero padding and performing DFT operations.

10 Short Question Answers:

1. Q: What is the relationship between the Z-transform and the DFT? A: The DFT is a special case of the Z-transform, specifically when the Z-transform is evaluated at points on the unit circle in the Z-plane, given by z = e^(j2πk/N).
   
   

2. Q: What does e^(j2πk/N) represent in the Z-plane? A: It represents points on the unit circle in the Z-plane, where N is the number of samples, and k ranges from 0 to N-1.
   
   

3. Q: What is an LTI system? A: An LTI system is a linear time-invariant system, characterized by its impulse response, h[n].
   
   

4. Q: How is the output of an LTI system calculated? A: The output, y[n], is calculated by the convolution of the input signal, x[n], with the system's impulse response, h[n]: y[n] = x[n] * h[n].
   
   

5. Q: What are the two types of convolution discussed in the context of DFT? A: The two types of convolution are linear convolution and circular convolution.
   
   

6. Q: What is the difference between linear and circular convolution? A: Linear convolution is the standard convolution, while circular convolution is what results when using the DFT.
   
   

7. Q: How is linear convolution converted to circular convolution for DFT analysis of LTI systems? A: Linear convolution is converted to circular convolution by padding the input and impulse response sequences with zeros to a length of N1 + N2 - 1, where N1 and N2 are the lengths of the input and impulse response, respectively.
   
   

8. Q: In DFT analysis, what operation is performed on the DFTs of the input and impulse response? A: The DFTs of the input and impulse response are multiplied to obtain the DFT of the output, Y[k] = X[k] * H[k].
   
   

9. Q: What is the purpose of taking the inverse DFT in LTI system analysis? A: The inverse DFT of Y[k] gives the output sequence y[n], which represents the linear convolution of x[n] and h[n].
   
   

10. Q: Given x[n] = {1, 2} and h[n] = {2, 1}, what is the result of the linear convolution? A: The linear convolution is y[n] = {2, 5, 2}.

Worked Out Examples:

The discrete Fourier transform (DFT) is a mathematical operation that transforms a discrete sequence of numbers into a sequence of complex numbers. The DFT is used in many applications, including signal processing, image processing, and communications. The DFT has a number of important properties, including: Linearity: The DFT of a linear combination of two sequences is the linear combination of the DFTs of the two sequences. Periodicity: The DFT of a periodic sequence is also periodic. Circular time shift: The DFT of a circularly time-shifted sequence is a circularly frequency-shifted version of the DFT of the original sequence. Time reversal: The DFT of a time-reversed sequence is the complex conjugate of the DFT of the original sequence. Conjugation: The DFT of the complex conjugate of a sequence is the complex conjugate of the DFT of the original sequence. Circular frequency shift: The DFT of a circularly frequency-shifted sequence is a circularly time-shifted version of the DFT of the original sequence. Multiplication: The DFT of the product of two sequences is the convolution of the DFTs of the two sequences. Circular convolution: The DFT of the circular convolution of two sequences is the product of the DFTs of the two sequences. Circular correlation: The DFT of the circular correlation of two sequences is the complex conjugate of the product of the DFTs of the two sequences. Parseval's relation: The sum of the squares of the magnitudes of the DFT coefficients of a sequence is equal to the energy of the sequence. These properties make the DFT a powerful tool for analyzing and manipulating discrete sequences of numbers. The DFT is often used in signal processing to analyze the frequency content of a signal. The DFT can also be used to filter signals, to remove noise, and to compress signals.

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