17.8:
Properties of DTFT II
Consider a discrete-time Fourier transform (DTFT) pair, differentiate both sides with respect to Ω, and then multiply by j.
The right-hand side becomes the Fourier transform of nx[n], illustrating the frequency differentiation property.
Apply discrete-time convolution to the DTFT pairs, change the order of the summations on the right-hand side, and apply the time-shifting property.
The substitution of the equations gives the time convolution property.
Multiply the two-time domain signals and interchange the order of integration and summation on the right-hand side.
This corresponds to periodic convolution in the frequency domain, scaled by the inverse of the period, demonstrating frequency convolution.
The accumulation property focuses on the summation of a discrete-time signal.
The DTFT of the accumulated signal has the original DTFT scaled by an exponential factor plus a delta function term, indicating periodic components at multiples of 2π.
The energy of the signal Ex is the sum of the squared magnitudes in the time domain. This is equal to the integral of the squared magnitudes of its DTFT, demonstrating Parseval's Relation.
17.8:
Properties of DTFT II
In the study of discrete-time signal processing, understanding the properties of the Discrete-Time Fourier Transform (DTFT) is crucial for analyzing and manipulating signals in the frequency domain. Several properties, including frequency differentiation, convolution, accumulation, and Parseval's relation, offer powerful tools for signal analysis.
The frequency differentiation property is illustrated by considering a DTFT pair and differentiating both sides with respect to ω. Multiplying by j (the imaginary unit), the right-hand side transforms into the Fourier transform of nx[n]. Mathematically, if
X(ejω) is the DTFT of x[n], then j(d/dω(X(ejω)) is the DTFT of nx[n]. This property is useful for finding the frequency characteristics related to the slope of the signal's spectrum.
When applying discrete-time convolution to DTFT pairs, we observe another important property. By changing the order of summation on the right-hand side and applying the time-shifting property, we arrive at the time convolution property. This states that the convolution of two signals in the time domain corresponds to the multiplication of their DTFTs in the frequency domain. Conversely, multiplying two signals in the time domain leads to a periodic convolution of their DTFTs in the frequency domain, scaled by the inverse of the period.
The accumulation property focuses on summing a discrete-time signal over time. The Discrete-Time Fourier Transform (DTFT) of this accumulated signal is related to the DTFT of the original signal but is modified by an exponential scaling factor. Additionally, there is a term that includes a delta function, which introduces periodic components at intervals of 2π in the frequency domain. This property highlights how accumulation in the time domain affects the frequency representation, leading to periodic features.
Parseval's Relation is a key result that links the energy of a signal in the time domain to its representation in the frequency domain. Specifically, the total energy of the signal
x[n], which is the sum of the squared magnitudes in the time domain, is equal to the integral of the squared magnitudes of its DTFT. This relation is fundamental in analyzing signal power and energy in both domains.
These properties collectively enhance the ability to analyze, design, and understand discrete-time systems, making them indispensable in digital signal processing.