The is a powerful tool that breaks down signals into their frequency components. It's like a musical ear that can pick out individual notes from a complex chord, allowing us to analyze and manipulate signals in ways we couldn't before.
This section dives into the math behind the Fourier transform and its key properties. We'll see how it relates to other transforms and learn to apply it to common functions, unlocking new ways to understand and work with signals.
Fourier transform definition and interpretation
Definition and mathematical representation
The Fourier transform decomposes a function into its constituent frequencies, representing the function in the
Defined as an integral of the form F(ω)=∫−∞∞f(t)e−jωtdt, where f(t) is the time-domain function, F(ω) is the frequency-domain function, and ω is the
The inverse Fourier transform recovers the original time-domain function from its frequency-domain representation, defined as f(t)=2π1∫−∞∞F(ω)ejωtdω
Complex-valued function and applications
The Fourier transform is a complex-valued function
Real part represents the amplitude of the frequency components
Imaginary part represents the phase of the frequency components
Widely used in various fields (signal processing, communications, control systems) to analyze and manipulate signals in the frequency domain
Properties of Fourier transforms
Linearity, scaling, and shifting properties
: The Fourier transform is a linear operation
Transform of a sum of functions equals the sum of their individual transforms
Transform of a scalar multiple of a function equals the scalar multiple of its transform
Scaling: If f(t) has a Fourier transform F(ω), then f(at) has a Fourier transform ∣a∣1F(aω), where a is a non-zero scalar
Relates the scaling of a function in the time domain to the scaling of its Fourier transform in the frequency domain
Time shifting: If f(t) has a Fourier transform F(ω), then f(t−t0) has a Fourier transform e−jωt0F(ω), where t0 is a real constant
A time shift in the time domain results in a phase shift in the frequency domain
: If f(t) has a Fourier transform F(ω), then ejω0tf(t) has a Fourier transform F(ω−ω0), where ω0 is a real constant
Multiplying a function by a complex exponential in the time domain results in a frequency shift in the frequency domain
Convolution and Parseval's theorem
Convolution: The convolution of two functions in the time domain is equivalent to the multiplication of their Fourier transforms in the frequency domain
If f(t) and g(t) have Fourier transforms F(ω) and G(ω), respectively, then the Fourier transform of their convolution, (f∗g)(t), is equal to F(ω)G(ω)
: Relates the energy of a function in the time domain to the energy of its Fourier transform in the frequency domain
The total energy of a signal is preserved when transformed between the time and frequency domains
Fourier transform evaluation
Fourier transforms of common functions
Rectangular pulse: The Fourier transform of a rectangular pulse with width τ and amplitude A is given by F(ω)=Aτsinc(2ωτ), where sinc(x)=xsin(x)
Gaussian pulse: The Fourier transform of a Gaussian pulse, f(t)=e−at2, is another Gaussian function, F(ω)=aπe−4aω2, where a>0
Signum function: The signum function, sgn(t), has a Fourier transform given by F(ω)=jω2, which is purely imaginary and has a singularity at ω=0
Inverse Fourier transforms of common functions
Rectangular pulse: The inverse Fourier transform of a rectangular pulse with width Ω and amplitude B is given by f(t)=2πBΩsinc(2Ωt)
Gaussian pulse: The inverse Fourier transform of a Gaussian pulse, F(ω)=e−aω2, is another Gaussian function, f(t)=4πa1e−4at2, where a>0
Fourier vs Laplace transforms
Relationship between Fourier and Laplace transforms
The Laplace transform is a generalization of the Fourier transform, extending the concept to complex frequencies (s=σ+jω) and causal signals (signals that are zero for t<0)
The Fourier transform can be considered a special case of the Laplace transform, where the real part of the complex frequency, σ, is set to zero
Definitions and applications
The Laplace transform is defined as L{f(t)}=F(s)=∫0∞f(t)e−stdt, while the Fourier transform is defined as F{f(t)}=F(ω)=∫−∞∞f(t)e−jωtdt
The Laplace transform is particularly useful for analyzing systems described by linear differential equations, as it allows for the conversion of differential equations into algebraic equations in the complex frequency domain
The Fourier transform is more suitable for analyzing the frequency content of signals and systems, while the Laplace transform is more appropriate for studying the stability and transient behavior of systems