Skip to main content

Fourier Transform (CTFT and DTFT)


Continuous Time Fourier Transform (CTFT)

Fourier transform is a process to convert a spatial domain signal (i.e., time domain signal) into a frequency domain signal. Oppositely, the inverse Fourier transform is a process to convert the frequency domain signal to the primary time domain signal.

Notation of CTFT

Let x(t) be a continuous-time signal. Then the CTFT is defined as:

\( X(j\omega) = \int_{-\infty}^{\infty} x(t) \cdot e^{-j\omega t} \, dt \)

Where:

  • \( \omega \) is the angular frequency in radians/second.
  • \( X(j\omega) \) is the frequency-domain representation of \( x(t) \).
  • The transform assumes signals are absolutely integrable over time.

Inverse CTFT:

\( x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} X(j\omega) \cdot e^{j\omega t} \, d\omega \)

 

Discrete Time Fourier Transform (DTFT)

The Discrete-Time Fourier Transform (DTFT) is used to analyze discrete-time signals, i.e., signals that are defined only at discrete intervals of time.

Notation of DTFT

Let x[n] be a discrete-time signal. Then the DTFT is defined as:

\( X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n] \cdot e^{-j\omega n} \)

Where:

  • \( \omega \) is the angular frequency in radians/sample.
  • \( X(e^{j\omega}) \) is periodic with period \( 2\pi \).

Inverse DTFT:

\( x[n] = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\omega}) \cdot e^{j\omega n} d\omega \)


Properties of Continuous-Time Fourier Transform (CTFT)

  • Linearity

    • The Fourier Transform is linear. For signals \(x_1(t)\) and \(x_2(t)\) with constants \(a_1, a_2\):

    \( \mathcal{F}\{a_1 x_1(t) + a_2 x_2(t)\} = a_1 X_1(j\omega) + a_2 X_2(j\omega) \)

  • Scaling

    • Time-scaling changes the width of the signal in time and inversely scales the frequency axis:

    \( \mathcal{F}\{x(at)\} = \frac{1}{|a|} X\left(j\frac{\omega}{a}\right) \)

  • Symmetry

    • Depending on whether the signal is real and even/odd:

    • Real and even: \(x(t) = x(-t) \Rightarrow X(j\omega) = X^*(-j\omega)\)
    • Real and odd: \(x(t) = -x(-t) \Rightarrow X(j\omega) = -X^*(-j\omega)\)

    Magnitude is always even, phase is odd for real signals.

  • Convolution

    • CTFT converts:

    • Time-domain convolution to frequency-domain multiplication: \( \mathcal{F}\{x_1(t) * x_2(t)\} = X_1(j\omega) \cdot X_2(j\omega) \)
    • Time-domain multiplication to frequency-domain convolution: \( \mathcal{F}\{x_1(t) \cdot x_2(t)\} = \frac{1}{2\pi} X_1(j\omega) * X_2(j\omega) \)

  • Shifting Property

    • Shifting in time introduces a linear phase shift in frequency domain:

    \( \mathcal{F}\{x(t - t_0)\} = e^{-j\omega t_0} X(j\omega) \)

  • Duality

    • If \( x(t) \leftrightarrow X(j\omega) \), then the roles of time and frequency can be interchanged:

    \( X(t) \leftrightarrow 2\pi x(-\omega) \)

  • Differentiation

    • Differentiating in time corresponds to multiplication by \(j\omega\) in frequency domain:

    \( \mathcal{F} \left\{ \frac{d}{dt}x(t) \right\} = j\omega X(j\omega) \)

  • Integration

    • Integration in time corresponds to division by \(j\omega\) in frequency domain (with a delta term if signal is not energy-limited):

    \( \mathcal{F} \left\{ \int_{-\infty}^{t} x(\tau)\, d\tau \right\} = \frac{X(j\omega)}{j\omega} + \pi X(0)\delta(\omega) \)

    In the Continuous-Time Fourier Transform (CTFT), X(0) represents the Fourier Transform of the signal at zero frequency, also called the DC component. It is defined as \( X(0) = \int_{-\infty}^{\infty} x(t)\, dt \) and corresponds to the total area under the time-domain signal.

  • Modulation Property

    • Multiplying by sine or cosine shifts frequency:

    \( \mathcal{F} \{ x(t) \cos(at) \} = \frac{1}{2} \left[ X(j(\omega + a)) + X(j(\omega - a)) \right] \)

    \( \mathcal{F} \{ x(t) \sin(at) \} = \frac{1}{2j} \left[ X(j(\omega + a)) - X(j(\omega - a)) \right] \)

  • Complex Conjugate Symmetry

    • If \(x(t)\) is complex, the Fourier Transform of the conjugate is:

    \( x^*(t) \longleftrightarrow X^*(-j\omega) \)

    For real signals, this gives conjugate symmetry:

    \( X(j\omega) = X^*(-j\omega) \Rightarrow |X(j\omega)| = |X(-j\omega)| \)

  • Parseval’s Theorem

    • Total energy in time equals total energy in frequency:

    \( \int_{-\infty}^{\infty} |x(t)|^2 \, dt = \frac{1}{2\pi} \int_{-\infty}^{\infty} |X(j\omega)|^2 \, d\omega \)

  • Time Reversal

    • Reversing the time signal flips the frequency axis:

    \( \mathcal{F}\{x(-t)\} = X(-j\omega) \)


    Properties of Discrete-Time Fourier Transform (DTFT)

  • Linearity

    • \( \mathcal{F}\{a_1 x_1[n] + a_2 x_2[n]\} = a_1 X_1(e^{j\omega}) + a_2 X_2(e^{j\omega}) \)

  • Time Shifting

    • \( \mathcal{F}\{x[n - n_0]\} = e^{-j\omega n_0} X(e^{j\omega}) \) – shifts introduce linear phase.

  • Frequency Shifting

    • \( \mathcal{F}\{x[n] e^{j\omega_0 n}\} = X(e^{j(\omega - \omega_0)}) \) – shifts in frequency domain.

  • Convolution

    • Time-domain convolution ↔ frequency-domain multiplication:

    \( \mathcal{F}\{x_1[n] * x_2[n]\} = X_1(e^{j\omega}) \cdot X_2(e^{j\omega}) \)

  • Frequency Domain Convolution

    • Multiplication in time ↔ convolution in frequency:

    \( \mathcal{F}\{x_1[n] \cdot x_2[n]\} = \frac{1}{2\pi} \int_{-\pi}^{\pi} X_1(e^{j\theta}) X_2(e^{j(\omega - \theta)}) d\theta \)

  • Symmetry

    • For Real and even \(x[n]\): \(X(e^{j\omega}) = X^*(e^{-j\omega})\), For Real and odd \(x[n]\): \(X(e^{j\omega}) = -X^*(e^{-j\omega})\)

  • Differencing

    • First difference in time: \(\mathcal{F}\{x[n]-x[n-1]\} = (1-e^{-j\omega})X(e^{j\omega})\)

  • Accumulation (Discrete Integration)

    • \(\mathcal{F}\{\sum_{k=-\infty}^n x[k]\} = \frac{X(e^{j\omega})}{1-e^{-j\omega}} + \pi X(e^{j0}) \delta(\omega)\)

  • Modulation Property

    • Multiplication by discrete cosine/sine:

    \(\mathcal{F}\{x[n]\cos(\omega_0 n)\} = \frac12[X(e^{j(\omega-\omega_0)}) + X(e^{j(\omega+\omega_0)})]\)

    \(\mathcal{F}\{x[n]\sin(\omega_0 n)\} = \frac{1}{2j}[X(e^{j(\omega-\omega_0)}) - X(e^{j(\omega+\omega_0)})]\)

  • Complex Conjugate Symmetry

    • For real \(x[n]\): \(X(e^{j\omega}) = X^*(e^{-j\omega})\) and \(|X(e^{j\omega})| = |X(e^{-j\omega})|\)

  • Parseval’s Theorem

    • \(\sum_{n=-\infty}^{\infty}|x[n]|^2 = \frac{1}{2\pi}\int_{-\pi}^{\pi}|X(e^{j\omega})|^2 d\omega\)

  • Time Reversal

    • \(\mathcal{F}\{x[-n]\} = X(e^{-j\omega})\)

  • Periodicity

    • DTFT is periodic with period \(2\pi\): \(X(e^{j\omega}) = X(e^{j(\omega+2\pi)})\)


    Fourier Transform of Some Common Signals (CTFT)

    • Fourier Transform of a Delta Function

      \( \delta(t) = \begin{cases} \infty & t = 0 \\ 0 & t \neq 0 \end{cases} \quad \text{(informal definition)} \)

      \( \int_{-\infty}^{+\infty} \delta(t)\, dt = 1 \)

      Thus Fourier transform of a delta/impulse is a constant equal to 1, independent of frequency:

      \( \mathcal{F}\{\delta(t)\} = 1 \)

      Dirac Delta Function

      Fig 1: Dirac Delta Function

      Fourier Transform of a Delta Function

      Fig 2: Fourier Transform of a Delta Function

    • Fourier Transform of a Unit Step Function

      u(t) = 0 for \( t < 0 \), and u(t) = 1 for \( t \ge 0 \)

      Its Fourier transform is given by:

      \( \mathcal{F}\{u(t)\} = \pi\delta(\omega) + \left(\frac{1}{j\omega}\right) \)

      Fourier Transform of a Unit Step Function

      Unit Step Function

      Fig 3: Unit Step Function

    • Fourier Transform of a Unit Pulse Function

      A pulse function can be represented as:

      \( x(t) = \Pi(t) = u(t + 1/2) - u(t - 1/2) \)

      For the rectangular pulse \( \text{rect}(t) = \Pi(t) \) defined as 1 for \( |t| \leq 1/2 \), and 0 otherwise:

      The Fourier transform of \( \text{rect}(t/\tau) \) is:

      \( \mathcal{F}\{\text{rect}(t/\tau)\} = \tau \frac{\sin(\omega \tau / 2)}{\omega \tau / 2} \)

      For \( \tau = 1 \),

      \( X(\omega) = \frac{\sin(\omega/2)}{\omega/2} = \text{sinc}(\omega/2) \)

      Unit Pulse

      Fig 4: Unit Pulse / Rectangular Pulse

      Fourier Transform of Rectangular Pulse

      Fig 5: Fourier Transform of the Continuous-Time Rectangular Pulse Π(t)

    • Fourier Transform of a Unit Triangle Pulse

      A unit triangle pulse is the convolution of a unit pulse with itself:

      \( \Lambda(t) = \Pi(t) * \Pi(t) \)

      \( \mathcal{F}\{\Lambda(t)\} = \text{sinc}^2(\omega/2) \)

      Unit Triangle Pulse

      Fig 6: Unit Triangle Pulse

      Fourier Transform of Unit Triangle Pulse

      Fig 7: Fourier Transform of the Continuous-Time Unit Triangle Pulse Λ(t)

    Common Discrete-Time Signals and their DTFT

    Note: DTFT is periodic with period \(2\pi\).

    • Unit Impulse Sequence

      Signal: \( x[n] = \delta[n] \)

      DTFT: \( X(e^{j\omega}) = 1 \)

      DTFT of Unit Impulse

      Fig 8: DTFT of the Discrete-Time Unit Impulse \(\delta[n]\)

    • Unit Step Sequence

      Signal: \( x[n] = u[n] \)

      DTFT: \( X(e^{j\omega}) = \pi \delta(\omega) + \left(\dfrac{1}{1 - e^{-j\omega}}\right) \)

      DTFT of Unit Step

      Fig 9: DTFT of the Discrete-Time Unit Step u[n]

    • Discrete-Time Sinusoid

      Signal: \( x[n] = \cos(\omega_0 n) \) or \( x[n] = \sin(\omega_0 n) \)

      DTFT: \( X(e^{j\omega}) = \pi \left[\delta(\omega - \omega_0) + \delta(\omega + \omega_0)\right] \)

      DTFT of Sinusoid

      Fig 10: DTFT of the Discrete-Time Sinusoid

    • Finite-Length Rectangular Pulse

      Signal (length \(N\), starting at \(n=0\)):

      \( x[n] = \begin{cases} 1, & 0 \le n \le N-1 \\ 0, & \text{otherwise} \end{cases} \)

      DTFT:

      \( X(e^{j\omega}) = e^{-j\omega\frac{N-1}{2}} \cdot \frac{\sin\left(\frac{N\omega}{2}\right)}{\sin\left(\frac{\omega}{2}\right)} \)

      Starting at \(n=n_0\) gives:

      \( X_{\text{shifted}}(e^{j\omega}) = e^{-j\omega n_0} \cdot e^{-j\omega\frac{N-1}{2}} \cdot \frac{\sin\left(\frac{N\omega}{2}\right)}{\sin\left(\frac{\omega}{2}\right)} \)

      DTFT of Finite-Length Rectangular Pulse

      Fig 11: DTFT of the Discrete-Time Finite-Length Rectangular Pulse

    Applications

    Fourier transform is used in circuit analysis, signal analysis, cell phones, image analysis, signal processing, and LTI systems. The Fourier transform is most probably the best tool to find the frequency in an entire field. This makes it a useful tool for LTI systems and signal processing. Partial differential equations reduce to ordinary differential equations in Fourier Transform.

    People are good at skipping over material they already know!

    View Related Topics to







    Contact Us

    Name

    Email *

    Message *

    Popular Posts

    BER vs SNR for M-ary QAM, M-ary PSK, QPSK, BPSK, ...(MATLAB Code + Simulator)

    📘 Overview of BER and SNR 🧮 Online Simulator for BER calculation 🧮 MATLAB Code for BER calculation 📚 Further Reading 📂 View Other Topics on M-ary QAM, M-ary PSK, QPSK ... 🧮 Online Simulator for Constellation Diagram of m-ary QAM 🧮 Online Simulator for Constellation Diagram of m-ary PSK 🧮 MATLAB Code for BER calculation of ASK, FSK, and PSK 🧮 MATLAB Code for BER calculation of Alamouti Scheme 🧮 Different approaches to calculate BER vs SNR What is Bit Error Rate (BER)? The abbreviation BER stands for Bit Error Rate, which indicates how many corrupted bits are received compared to the total number of bits sent. BER = (number of bits received in error) / (total number of transmitted bits) What is Signal-to-Noise Ratio (SNR)? SNR is the ratio of signal power to noise powe...
    Read more

    ASK, FSK, and PSK (with MATLAB + Online Simulator)

    📘 Overview 📘 Amplitude Shift Keying (ASK) 📘 Frequency Shift Keying (FSK) 📘 Phase Shift Keying (PSK) 📘 Which of the modulation techniques—ASK, FSK, or PSK—can achieve higher bit rates? 🧮 MATLAB Codes 📘 Simulator for binary ASK, FSK, and PSK Modulation 📚 Further Reading ASK or OFF ON Keying ASK is a simple (less complex) Digital Modulation Scheme where we vary the modulation signal's amplitude or voltage by the message signal's amplitude or voltage. We select two levels (two different voltage levels) for transmitting modulated message signals. For example, "+5 Volt" (upper level) and "0 Volt" (lower level). To transmit binary bit "1", the transmitter sends "+5 Volts", and for bit "0", it sends no power. The receiver uses filters to detect whether a binary "1" or "0" was transmitted. ...
    Read more

    Calculation of SNR from FFT bins in MATLAB

    📘 Overview 🧮 MATLAB Code for Estimation of SNR from FFT bins 🧮 MATLAB Code for SNR from PSD using Kaiser Window 📚 Further Reading Here, you can find the SNR of a received signal from periodogram / FFT bins using the Kaiser operator. The beta (β) parameter characterizes the Kaiser window, which controls the trade-off between the main lobe width and the side lobe level. Steps Set up the sampling rate and time vector Compute the FFT and periodogram Calculate the frequency resolution and signal power Exclude the signal power from noise calculation Compute the noise power and SNR MATLAB Code for Estimation of SNR from FFT bins clc; clear; close all; % Parameters fs = 8000; f_tone = 1000; N = 8192; t = (0:N-1)/fs; % Generate signal + noise signal = sin(2*pi*f_tone*t); SNR_true_dB = 20; signal_power = mean(signal.^2); noise_power = signal_power / (10^(SNR_true_dB/10)); noisy_signal = signal + sqrt(noise_power) * randn(1, N); % Apply ...
    Read more

    MATLAB Code for ASK, FSK, and PSK (with Online Simulator)

    📘 Overview & Theory 🧮 MATLAB Code for ASK 🧮 MATLAB Code for FSK 🧮 MATLAB Code for PSK 🧮 Simulator for binary ASK, FSK, and PSK Modulations 📚 Further Reading ASK, FSK & PSK HomePage MATLAB Code MATLAB Code for ASK Modulation and Demodulation % The code is written by SalimWireless.Com % Clear previous data and plots clc; clear all; close all; % Parameters Tb = 1; % Bit duration (s) fc = 10; % Carrier frequency (Hz) N_bits = 10; % Number of bits Fs = 100 * fc; % Sampling frequency (ensure at least 2*fc, more for better representation) Ts = 1/Fs; % Sampling interval samples_per_bit = Fs * Tb; % Number of samples per bit duration % Generate random binary data rng(10); % Set random seed for reproducibility binary_data = randi([0, 1], 1, N_bits); % Generate random binary data (0 or 1) % Initialize arrays for continuous signals t_overall = 0:Ts:(N_bits...
    Read more

    Online Simulator for ASK, FSK, and PSK

    Try our new Digital Signal Processing Simulator!   Start Simulator for binary ASK Modulation Message Bits (e.g. 1,0,1,0) Carrier Frequency (Hz) Sampling Frequency (Hz) Run Simulation Simulator for binary FSK Modulation Input Bits (e.g. 1,0,1,0) Freq for '1' (Hz) Freq for '0' (Hz) Sampling Rate (Hz) Visualize FSK Signal Simulator for BPSK Modulation ...
    Read more

    Comparing Baseband and Passband Implementations of ASK, FSK, and PSK

    📘 Overview 🧮 Baseband and Passband Implementations of ASK, FSK, and PSK 🧮 Difference betwen baseband and passband 📚 Further Reading 📂 Other Topics on Baseband and Passband ... 🧮 Baseband modulation techniques 🧮 Passband modulation techniques   Baseband modulation techniques are methods used to encode information signals onto a baseband signal (a signal with frequencies close to zero). Passband techniques shift these signals to higher carrier frequencies for transmission. Here are the common implementations: Amplitude Shift Keying (ASK) [↗] : In ASK, the amplitude of the signal is varied to represent different symbols. Binary ASK (BASK) is a common implementation where two different amplitudes represent binary values (0 and 1). ASK is simple but susceptible to noise. ASK Baseband (Digital Bits) ASK Passband (Modulated Carrier)     Fig 1:  ASK Passband Modulation (...
    Read more

    Comparing Baseband and Passband Implementations of m-ary QAM

      Let's assume your original digital message bitstream is: 0, 0, 1, 0, 0, 0, 1, 0, 1, 1 In 4-QAM, we group them into pairs: (00), (10), (00), (10), (11). Your baseband symbols are: Symbol 1 (Bits 00): -1.00 - j1.00 Symbol 2 (Bits 10): 1.00 - j1.00 Symbol 3 (Bits 00): -1.00 - j1.00 Symbol 4 (Bits 10): 1.00 - j1.00 Symbol 5 (Bits 11): 1.00 + j1.00   To transmit these symbols over a wireless medium, we modulate this baseband signal onto a high-frequency carrier (e.g., 50 Hz). This process creates the passband signal , where the information is stored in the phase and amplitude of the sine wave. Fig 1: 4-QAM Baseband I and Q Components Fig 2: 4-QAM Passband Modulated Signal   In this example, the symbol rate is 5 symbols per second. Detailed Explanation 4-QAM Constellation Mapping In standard 4-QAM mapping, bits are converted to complex points on a grid: Bits...
    Read more

    Constellation Diagrams of ASK, PSK, and FSK with MATLAB Code + Simulator

    📘 Overview of Energy per Bit (Eb / N0) 🧮 Online Simulator for constellation diagrams of ASK, FSK, and PSK 🧮 Theory behind Constellation Diagrams of ASK, FSK, and PSK 🧮 MATLAB Codes for Constellation Diagrams of ASK, FSK, and PSK 📚 Further Reading 📂 Other Topics on Constellation Diagrams of ASK, PSK, and FSK ... 🧮 Simulator for constellation diagrams of m-ary PSK 🧮 Simulator for constellation diagrams of m-ary QAM BASK (Binary ASK) Modulation: Transmits one of two signals: 0 or -√Eb, where Eb​ is the energy per bit. These signals represent binary 0 and 1.    BFSK (Binary FSK) Modulation: Transmits one of two signals: +√Eb​ ( On the y-axis, the phase shift of 90 degrees with respect to the x-axis, which is also termed phase offset ) or √Eb (on x-axis), where Eb​ is the energy per bit. These signals represent binary 0 and 1.  BPSK (Binary PSK) Modulation: Transmits one of two signals...
    Read more