4.2 Scaling and Duality

Scaling and duality are two key properties of the Fourier transform that help simplify calculations and reveal symmetries between time and frequency domains. Scaling shows how stretching or compressing a signal affects its frequency content, while duality allows us to swap time and frequency roles.

These properties are super useful for understanding how signals behave when manipulated. They let us quickly figure out Fourier transforms of modified signals without starting from scratch, saving time and effort in signal processing applications.

Scaling Property of Fourier Transform

Time Domain Scaling and Frequency Domain Effects

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• The scaling property states that if a signal is scaled in the time domain by a factor of $a$, its Fourier transform will be scaled in the frequency domain by a factor of $1/a$ and its amplitude will be scaled by a factor of $1/|a|$
• Scaling a signal in the time domain results in an inverse scaling in the frequency domain, affecting both the frequency and amplitude of the Fourier transform
  • If a signal is compressed in the time domain ($a > 1$), its Fourier transform will be expanded in the frequency domain and its amplitude will be decreased by a factor of $1/a$
  • If a signal is expanded in the time domain ($a < 1$), its Fourier transform will be compressed in the frequency domain and its amplitude will be increased by a factor of $1/a$
  • Example: If a signal $x(t)$ is scaled by a factor of 2 to become $x(2t)$, its Fourier transform $X(\omega)$ will be scaled to become $(1/2)X(\omega/2)$

Mathematical Representation and Fourier Transform Pairs

• The scaling property can be represented mathematically as: if $x(at) \leftrightarrow X(\omega)$, then $x(t) \leftrightarrow (1/|a|)X(\omega/a)$, where $\leftrightarrow$ denotes the Fourier transform pair
• The scaling factor $a$ in the time domain corresponds to the reciprocal scaling factor $1/a$ in the frequency domain
• The amplitude scaling factor $1/|a|$ ensures that the energy of the signal is preserved under scaling
• Example: If the Fourier transform pair is known for a signal $x(t) \leftrightarrow X(\omega)$, the Fourier transform of the scaled signal $x(2t)$ can be directly determined as $(1/2)X(\omega/2)$ using the scaling property

Time vs Frequency Domain Duality

Symmetrical Relationship and Interchangeable Roles

• The duality property states that if a signal has a Fourier transform pair $x(t) \leftrightarrow X(\omega)$, then its dual Fourier transform pair is $X(t) \leftrightarrow 2\pi x(-\omega)$
• Duality establishes a symmetrical relationship between the time and frequency domains, allowing the roles of time and frequency to be interchanged in Fourier transform pairs
  • If a signal is symmetric in the time domain, its Fourier transform will be symmetric in the frequency domain, and vice versa
  • Example: If $x(t)$ is an even function, then its Fourier transform $X(\omega)$ will also be an even function

Deriving Fourier Transforms using Duality

• The duality property can be used to derive the Fourier transform of a signal if the Fourier transform of its dual signal is known
• By interchanging the roles of time and frequency and applying the appropriate scaling factor, the Fourier transform of a dual signal can be determined
• The factor of $2\pi$ in the duality property arises from the definition of the Fourier transform and its inverse
• Example: If the Fourier transform of a rectangular pulse $\text{rect}(t)$ is known to be $\text{sinc}(\omega)$, the Fourier transform of $\text{sinc}(t)$ can be derived using duality as $2\pi \text{rect}(-\omega)$

Simplifying Fourier Transform Calculations

Applying Scaling Property for Simplified Calculations

• The scaling property can be applied to simplify Fourier transform calculations by relating the Fourier transform of a scaled signal to the Fourier transform of the original signal
• When a signal is scaled in the time domain, the scaling property can be used to determine the corresponding scaling in the frequency domain, eliminating the need to calculate the Fourier transform directly
• Example: If the Fourier transform of a Gaussian function $e^{-\pi t^2}$ is known to be $e^{-\pi \omega^2}$, the Fourier transform of a scaled Gaussian function $e^{-\pi (at)^2}$ can be determined using the scaling property as $(1/|a|)e^{-\pi (\omega/a)^2}$

Employing Duality Property for Fourier Transform Derivation

• The duality property can be employed to determine the Fourier transform of a signal by considering the Fourier transform of its dual signal, which may be easier to calculate or already known
• By recognizing the dual relationship between time and frequency domains, the Fourier transform of a signal can be derived from its dual counterpart
• Example: The Fourier transform of a triangular function can be derived using the duality property and the known Fourier transform of a sinc-squared function

Combining Scaling and Duality for Complex Signals

• Combining the scaling and duality properties allows for the derivation of Fourier transform pairs for scaled and dual signals, reducing the complexity of Fourier transform calculations
• By recognizing patterns and symmetries in signals and their Fourier transforms, the scaling and duality properties can be used to simplify the analysis and computation of Fourier transforms in various applications
• Example: The Fourier transform of a chirp signal $e^{j\pi at^2}$ can be determined by applying the scaling property to the known Fourier transform of a Gaussian function and then using the duality property to obtain the final result