1.2 Continuous-time Fourier transform

The continuous-time Fourier transform (CTFT) is a powerful tool for analyzing signals in the frequency domain. It decomposes continuous-time signals into their frequency components, providing insights into spectral content and properties. This mathematical technique is essential for understanding signal behavior and designing effective processing systems.

CTFT forms the foundation for advanced signal processing techniques. It enables the study of signal characteristics, system responses, and transformations between time and frequency domains. Understanding CTFT properties, basic signal transforms, and relationships with other transforms is crucial for mastering signal analysis and manipulation.

Definition of continuous-time Fourier transform

• The continuous-time Fourier transform (CTFT) is a mathematical tool used to analyze and represent continuous-time signals in the frequency domain
• It decomposes a signal into its constituent frequency components, providing insights into the signal's spectral content and properties
• The CTFT is widely used in advanced signal processing to study the frequency characteristics of signals and to design and analyze systems

Mathematical representation

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• The CTFT of a continuous-time signal $x(t)$ is denoted as $X(f)$ and is defined by the integral: $X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} dt$
  • $f$ represents the frequency variable in Hertz (Hz)
  • $j$ is the imaginary unit, defined as $j^2 = -1$
• The CTFT maps the time-domain signal $x(t)$ to its frequency-domain representation $X(f)$, which is a complex-valued function of frequency

Fourier transform pair

• The CTFT and the original time-domain signal form a Fourier transform pair, denoted as $x(t) \leftrightarrow X(f)$
• This pair highlights the duality between the time and frequency domains, indicating that a signal can be uniquely represented in either domain
• The arrow notation $\leftrightarrow$ signifies that the CTFT is a bijective mapping between the time and frequency domains

Inverse Fourier transform

• The inverse CTFT allows the recovery of the time-domain signal $x(t)$ from its frequency-domain representation $X(f)$
• The inverse CTFT is defined by the integral: $x(t) = \int_{-\infty}^{\infty} X(f) e^{j2\pi ft} df$
• The inverse CTFT completes the bidirectional relationship between the time and frequency domains, enabling seamless conversion between the two representations

Properties of Fourier transform

• The CTFT exhibits several important properties that facilitate the analysis and manipulation of signals in the frequency domain
• These properties provide valuable insights into the behavior of signals under various operations and transformations
• Understanding and leveraging these properties is crucial for effective signal processing and system design

Linearity

• The CTFT is a linear operation, meaning that it satisfies the principles of superposition and scaling
• If $x_1(t) \leftrightarrow X_1(f)$ and $x_2(t) \leftrightarrow X_2(f)$, then:
  • Superposition: $a_1x_1(t) + a_2x_2(t) \leftrightarrow a_1X_1(f) + a_2X_2(f)$
  • Scaling: $ax(t) \leftrightarrow aX(f)$, where $a$ is a constant
• Linearity allows the CTFT to be applied to linear combinations of signals and enables the analysis of complex signals by decomposing them into simpler components

Time shifting

• The CTFT of a time-shifted signal $x(t-t_0)$ is given by: $x(t-t_0) \leftrightarrow X(f)e^{-j2\pi ft_0}$
• Time shifting in the time domain introduces a linear phase shift in the frequency domain, with the phase shift proportional to the time delay $t_0$
• This property is useful for understanding the effect of time delays on signals and systems

Frequency shifting

• The CTFT of a frequency-shifted signal $x(t)e^{j2\pi f_0t}$ is given by: $x(t)e^{j2\pi f_0t} \leftrightarrow X(f-f_0)$
• Frequency shifting in the time domain corresponds to a translation of the spectrum in the frequency domain by $f_0$
• This property is fundamental to the concept of modulation, where a signal is shifted in frequency to transmit information

Time scaling

• The CTFT of a time-scaled signal $x(at)$, where $a$ is a non-zero constant, is given by: $x(at) \leftrightarrow \frac{1}{|a|}X(\frac{f}{a})$
• Time scaling in the time domain results in an inverse scaling of the frequency axis and a magnitude scaling of the spectrum
• This property is relevant in applications such as signal compression and expansion

Duality

• The CTFT exhibits a duality property, which states that if $x(t) \leftrightarrow X(f)$, then $X(t) \leftrightarrow x(-f)$
• The duality property highlights the symmetry between the time and frequency domains, indicating that the roles of time and frequency can be interchanged
• This property is exploited in various signal processing techniques, such as the design of dual filter banks

Parseval's theorem

• Parseval's theorem relates the energy of a signal in the time domain to its energy in the frequency domain
• It states that the total energy of a signal $x(t)$ is equal to the total energy of its Fourier transform $X(f)$:
  • $\int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df$
• Parseval's theorem provides a powerful tool for analyzing the energy distribution of signals across different frequency components

Fourier transform of basic signals

• The CTFT can be applied to various basic signals to obtain their frequency-domain representations
• These basic signals serve as building blocks for more complex signals and provide insights into the spectral characteristics of different waveforms
• Understanding the Fourier transforms of basic signals is essential for analyzing and interpreting the frequency content of real-world signals

Impulse function

• The impulse function, also known as the Dirac delta function $\delta(t)$, is a fundamental signal in signal processing
• The CTFT of the impulse function is given by: $\delta(t) \leftrightarrow 1$
• The impulse function has a constant spectrum of unity across all frequencies, indicating that it contains equal energy at all frequencies
• The impulse function is used to model instantaneous events and is a key component in the analysis of linear time-invariant (LTI) systems

Step function

• The unit step function, denoted as $u(t)$, represents a signal that transitions from zero to one at time $t=0$
• The CTFT of the unit step function is given by: $u(t) \leftrightarrow \frac{1}{j2\pi f} + \frac{1}{2}\delta(f)$
• The spectrum of the unit step function consists of a constant term and an impulse at zero frequency, indicating the presence of a DC component
• The step function is commonly used to model abrupt changes or transitions in signals

Rectangular pulse

• The rectangular pulse, also known as the rect function, is defined as a signal that is unity over a specific time interval and zero elsewhere
• The CTFT of a rectangular pulse with duration $T$ is given by: $\text{rect}(\frac{t}{T}) \leftrightarrow T\text{sinc}(fT)$, where $\text{sinc}(x) = \frac{\sin(\pi x)}{\pi x}$
• The spectrum of a rectangular pulse is a sinc function, with the main lobe width inversely proportional to the pulse duration $T$
• Rectangular pulses are used in various applications, such as pulse shaping in communication systems and windowing in spectral analysis

Triangular pulse

• The triangular pulse is a signal that linearly ramps up to a peak value and then linearly ramps down to zero
• The CTFT of a triangular pulse with duration $T$ is given by: $\Lambda(\frac{t}{T}) \leftrightarrow T\text{sinc}^2(\frac{fT}{2})$, where $\Lambda(t)$ is the triangular function
• The spectrum of a triangular pulse is the squared sinc function, exhibiting a faster decay compared to the rectangular pulse
• Triangular pulses are used in applications such as signal interpolation and anti-aliasing filtering

Sinusoidal signal

• A sinusoidal signal, also known as a pure tone, is a fundamental waveform in signal processing
• The CTFT of a sinusoidal signal $\cos(2\pi f_0t)$ is given by: $\cos(2\pi f_0t) \leftrightarrow \frac{1}{2}[\delta(f-f_0) + \delta(f+f_0)]$
• The spectrum of a sinusoidal signal consists of two impulses located at the positive and negative frequencies $\pm f_0$, indicating the presence of a single frequency component
• Sinusoidal signals are the basis for the analysis and synthesis of more complex signals using the Fourier series and Fourier transform techniques

Fourier transform of periodic signals

• Periodic signals are signals that repeat themselves at regular intervals, with a fixed period $T$
• The CTFT of periodic signals is closely related to the Fourier series representation, which decomposes a periodic signal into a sum of sinusoidal components
• Understanding the relationship between the Fourier series and the Fourier transform is crucial for analyzing and processing periodic signals in various domains

Fourier series representation

• The Fourier series represents a periodic signal $x(t)$ with period $T$ as an infinite sum of sinusoidal components:
  • $x(t) = \sum_{n=-\infty}^{\infty} c_n e^{j2\pi nf_0t}$, where $f_0 = \frac{1}{T}$ is the fundamental frequency and $c_n$ are the complex Fourier series coefficients
• The Fourier series coefficients $c_n$ are given by: $c_n = \frac{1}{T} \int_{0}^{T} x(t) e^{-j2\pi nf_0t} dt$
• The Fourier series representation allows for the analysis and synthesis of periodic signals in terms of their harmonic components
• The coefficients $c_n$ provide information about the amplitude and phase of each harmonic component

Relationship between Fourier series and Fourier transform

• The CTFT of a periodic signal $x(t)$ with period $T$ is given by: $x(t) \leftrightarrow X(f) = \sum_{n=-\infty}^{\infty} c_n \delta(f - nf_0)$
• The Fourier transform of a periodic signal consists of a series of impulses located at integer multiples of the fundamental frequency $f_0$, with the impulse amplitudes equal to the Fourier series coefficients $c_n$
• The relationship between the Fourier series and the Fourier transform highlights the spectral nature of periodic signals, where the energy is concentrated at discrete frequencies
• This relationship allows for the seamless transition between the time-domain representation (Fourier series) and the frequency-domain representation (Fourier transform) of periodic signals

Convolution and multiplication properties

• Convolution and multiplication are two fundamental operations in signal processing that have important implications in the time and frequency domains
• The CTFT provides a powerful framework for analyzing the effects of convolution and multiplication on signals and systems
• Understanding these properties is essential for designing and implementing signal processing algorithms and systems

Convolution in time domain

• The convolution of two continuous-time signals $x(t)$ and $h(t)$ is denoted as $x(t) * h(t)$ and is defined by the integral:
  • $(x * h)(t) = \int_{-\infty}^{\infty} x(\tau)h(t-\tau) d\tau$
• Convolution in the time domain represents the interaction between two signals, where one signal is shifted and scaled by the other signal at each time instant
• Convolution is used to model the output of a linear time-invariant (LTI) system when an input signal is applied, with $h(t)$ representing the system's impulse response

Multiplication in frequency domain

• The CTFT of the convolution of two signals $x(t)$ and $h(t)$ is given by the multiplication of their individual Fourier transforms:
  • $x(t) * h(t) \leftrightarrow X(f)H(f)$
• Multiplication in the frequency domain corresponds to convolution in the time domain, providing a powerful duality between the two operations
• This property simplifies the analysis and design of LTI systems in the frequency domain, as the system's frequency response can be obtained by multiplying the input signal's spectrum with the system's transfer function

Applications in signal processing

• The convolution and multiplication properties of the CTFT have numerous applications in signal processing, including:
  • Filtering: Convolution in the time domain is equivalent to multiplication in the frequency domain, allowing for the design and implementation of various filters (low-pass, high-pass, band-pass, etc.)
  • Modulation: Multiplication of signals in the time domain results in convolution of their spectra in the frequency domain, which is the basis for amplitude modulation (AM) and frequency modulation (FM) techniques
  • Deconvolution: The inverse operation of convolution, deconvolution aims to recover the original signal from a convolved signal, and it is used in applications such as signal restoration and echo cancellation

Relationship with Laplace transform

• The Laplace transform is a generalization of the CTFT that extends the concept of frequency-domain analysis to complex frequencies
• While the CTFT deals with real-valued frequencies, the Laplace transform introduces a complex frequency variable $s = \sigma + j\omega$, where $\sigma$ represents the real part and $\omega$ represents the imaginary part
• Understanding the relationship between the CTFT and the Laplace transform is important for analyzing and designing systems with more complex behaviors, such as stability and causality

Similarities and differences

• The Laplace transform shares many similarities with the CTFT, including linearity, time shifting, and convolution properties
• However, the Laplace transform differs from the CTFT in several key aspects:
  • The Laplace transform is defined for complex frequencies, while the CTFT is defined for real frequencies
  • The Laplace transform can handle signals that are not absolutely integrable, while the CTFT requires absolute integrability
  • The Laplace transform provides additional information about the stability and causality of a system, which is not directly available in the CTFT

Regions of convergence

• The region of convergence (ROC) is a crucial concept in the Laplace transform that determines the range of complex frequencies for which the Laplace transform converges
• The ROC provides information about the stability and causality of a system based on the location of its poles and zeros in the complex plane
• The CTFT can be considered a special case of the Laplace transform, where the ROC includes the imaginary axis ($\sigma = 0$)

Stability analysis

• The Laplace transform allows for the stability analysis of systems by examining the location of the system's poles in the complex plane
• A system is considered stable if all its poles lie in the left half-plane ($\sigma < 0$), indicating that the system's response decays over time
• The CTFT alone does not provide direct information about system stability, as it only considers the frequency response on the imaginary axis
• The Laplace transform extends the stability analysis capabilities of the CTFT by incorporating the complex frequency domain

Sampling and aliasing

• Sampling is the process of converting a continuous-time signal into a discrete-time signal by capturing its values at regular time intervals
• Aliasing is a phenomenon that occurs when a continuous-time signal is sampled at an insufficient rate, leading to the distortion or misinterpretation of the signal's frequency content
• Understanding sampling and aliasing is crucial for the proper digitization and processing of continuous-time signals in practical applications

Nyquist sampling theorem

• The Nyquist sampling theorem, also known as the Shannon sampling theorem, establishes the minimum sampling rate required to avoid aliasing and accurately reconstruct a band-limited signal
• The theorem states that a continuous-time signal with a maximum frequency component $f_{max}$ can be perfectly reconstructed from its samples if the sampling rate $f_s$ satisfies: $f_s > 2f_{max}$
• The minimum sampling rate $f_s = 2f_{max}$ is called the Nyquist rate, and it ensures that the original signal can be recovered without any loss of information

Aliasing in frequency domain

• Aliasing occurs when the sampling rate is lower than the Nyquist rate, causing high-frequency components of the signal to be misinterpreted as low-frequency components
• In the frequency domain, aliasing manifests as the overlapping of the signal's spectrum, where frequency components above the Nyquist frequency ($f_s/2$) are folded back into the lower frequency range
• Aliasing distorts the signal's spectrum and can lead to the loss of information and the introduction of artifacts in the reconstructed signal

Anti-aliasing filters

• Anti-aliasing filters