DTFT and Z-Transform: Key Concepts and
Properties
This document highlights important concepts, properties, and an exam question
analysis related to the Discrete-Time Fourier Transform (DTFT) and the Z-Transform,
based on the provided learning material.
Key Concepts
1. Discrete-Time Fourier Transform (DTFT):
Definition: The DTFT transforms a discrete-time signal (a sequence of
numbers) into its frequency-domain representation. This representation,
X(e^jω) , is a continuous and periodic function of frequency.
Purpose: It allows us to analyze the frequency components (like sine and
cosine waves) that constitute a discrete signal.
Inverse DTFT: This operation converts the frequency-domain
representation back into the time-domain signal.
2. Z-Transform:
Definition: The Z-transform is a generalization of the DTFT, converting a
discrete-time signal x[n] into a function X(z) of a complex variable z .
Relationship to DTFT: The DTFT is a specific instance of the Z-transform
evaluated on the "unit circle" (where the magnitude of z is 1).
Region of Convergence (ROC): This is the set of values for z for which the
Z-transform converges. The ROC is critical for determining system
properties such as stability.
Stability: For a causal Linear Time-Invariant (LTI) system, stability is
ensured if all the poles of its transfer function lie inside the unit circle.
3. Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT):
The DTFT is a theoretical tool that yields a continuous function, which is not
directly computable. The DFT is its practical, computable counterpart,
operating on finite-length signals and producing a discrete set of frequency
samples.
The FFT is an highly efficient algorithm specifically designed for calculating
the DFT.
Important Properties
The following table summarizes several fundamental properties crucial for working
with these transforms, enabling efficient problem-solving.
Property
Linearity
Explanation
Importance
The transform of a weighted sum of
Simplifies the analysis of complex
signals is the same weighted sum of
signals by breaking them down into
their individual transforms.
simpler parts.
Shifting a signal in the time domain
Time Shifting
by n₀ results in multiplying its
DTFT by a complex exponential
e^(-jωn₀) .
Frequency
Shifting
Multiplying a time-domain signal by
a complex exponential e^(jω₀n)
shifts its frequency spectrum.
Reversing a signal in time ( x[-n] )
Time Reversal
reflects its frequency spectrum
around the vertical axis.
Convolution
Relates time delays directly to
phase changes in the frequency
domain.
This is the mathematical basis for
modulation.
Shows the effect of playing a signal
backward.
(Crucial Property) Convolution of
This turns a very complex
two signals in the time domain is
calculation (convolution) into a
equivalent to the multiplication of
simple multiplication, which is
their transforms in the frequency
often the easiest way to find the
domain.
output of an LTI system.
The multiplication of two signals in
Multiplication
the time domain corresponds to the
This is the dual property of
convolution of their transforms in
convolution.
the frequency domain.
Exam Question Analysis
Question: "If X(n) is given as delta functions and you have another Y(n) or another
second signal given as a U(n). So you have a impulse signal and you have a step signal.
How you will perform the convolution or shifting with them?"
This question assesses your understanding of both the convolution property and the
sifting property of the delta function.
Detailed Explanation:
Consider the convolution of an impulse signal δ[n-n₀] (an impulse shifted by n₀ )
with a step signal u[n] .
The convolution sum is defined as: y[n] = x[n] * h[n] = Σ x[k]h[n-k] (sum over
all k)
Let x[n] = u[n] and h[n] = δ[n-n₀] .
1. Using the Sifting Property of the Delta Function: The fundamental property of
the delta function δ[n-n₀] is that it is zero everywhere except at n = n₀ . When
multiplied by another function f[n] and summed, it effectively "sifts" out the
value of f[n] at the point n₀ .
Applying this to the convolution sum: y[n] = u[n] * δ[n-n₀] = Σ u[k]δ[n-kn₀]
The delta function δ[n-k-n₀] is non-zero only when its argument is zero, i.e., nk-n₀ = 0 , which implies k = n-n₀ . Consequently, only the term where k is
replaced by n-n₀ survives the summation.
The result is: y[n] = u[n-n₀]
Conclusion: Convolving any signal with a shifted impulse δ[n-n₀] results in a
corresponding shift of the original signal by the same amount. This is a
foundational identity in signal processing.
2. Using the Frequency Domain (DTFT): Alternatively, the convolution property
can be leveraged:
The DTFT of u[n] is U(e^jω) .
The DTFT of δ[n-n₀] is e^(-jωn₀) .
The DTFT of the output Y(e^jω) is the product of the individual DTFTs: Y(e^jω)
= U(e^jω) * e^(-jωn₀)
Based on the time-shifting property, multiplying a transform by e^(-jωn₀) is
equivalent to shifting the original time-domain signal by n₀ . Therefore, the
inverse DTFT of this expression is u[n-n₀] . Both methods consistently yield the
same correct result.
Solving Question: Convolution of Impulse and Step Signals
Question: Given two discrete-time signals: 1. An impulse signal: x[n] = δ[n-2] 2. A
step signal: h[n] = u[n]
Determine the output signal y[n] when x[n] is convolved with h[n] , i.e., y[n] =
x[n] * h[n] . Explain your steps using both the sifting property of the delta function
and the convolution property in the frequency domain.
Solution:
We need to find y[n] = δ[n-2] * u[n] .
Method 1: Using the Sifting Property of the Delta Function
The convolution sum is given by: y[n] = Σ x[k]h[n-k] (sum over all k from -∞ to
+∞)
Substitute x[k] = δ[k-2] and h[n-k] = u[n-k] into the convolution sum: y[n] =
Σ δ[k-2]u[n-k]
The sifting property of the discrete-time impulse function δ[k-k₀] states that Σ δ[kk₀]f[k] = f[k₀] . In our case, k₀ = 2 and f[k] = u[n-k] .
Applying the sifting property, the sum becomes non-zero only when k = 2 : y[n] =
u[n-2]
Explanation: Convolving any signal with a shifted impulse δ[n-k₀] simply shifts the
original signal by k₀ . Here, the step signal u[n] is shifted by 2 units to the right,
resulting in u[n-2] .
Method 2: Using the Convolution Property in the Frequency Domain
The convolution property states that convolution in the time domain corresponds to
multiplication in the frequency domain. That is, if y[n] = x[n] * h[n] , then
Y(e^jω) = X(e^jω) * H(e^jω) .
First, find the DTFT of each signal: 1. DTFT of x[n] = δ[n-2] : Using the time-shifting
property, the DTFT of δ[n-k₀] is e^(-jωk₀) . Therefore, X(e^jω) = e^(-jω2)
1. DTFT of h[n] = u[n] : The DTFT of the unit step function u[n] is: H(e^jω) = 1
/ (1 - e^(-jω)) + π Σ δ(ω - 2πk) (for k = 0, ±1, ±2, ...) For practical
purposes in convolution, we often consider the principal value or focus on the
non-impulse part, but for this specific problem, the time-shifting property is
more direct.
Now, multiply their DTFTs: Y(e^jω) = X(e^jω) * H(e^jω) = e^(-jω2) * H(e^jω)
Finally, take the Inverse DTFT of Y(e^jω) : We know that if H(e^jω) is the DTFT of
h[n] , then e^(-jωk₀)H(e^jω) is the DTFT of h[n-k₀] . Here, k₀ = 2 and h[n] =
u[n] .
Therefore, the inverse DTFT of e^(-jω2)H(e^jω) is u[n-2] .
Conclusion: Both methods confirm that the convolution of δ[n-2] with u[n] results
in u[n-2] . This demonstrates the power and consistency of using both time-domain
properties and frequency-domain analysis to solve signal processing problems.