The discrete-time Fourier transform (DTFT) is a mathematical representation of a discrete-time aperiodic signal as a continuous function of frequency, expressed as an infinite sum of complex exponentials.[1] It is defined by the analysisequation X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x e^{-j \omega n}, where x is the discrete-time signal, \omega is the angular frequency, and the transform X(e^{j\omega}) is periodic with period $2\pi.[1] The inverse DTFT, or synthesis equation, recovers the original signal via x = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\omega}) e^{j \omega n} \, d\omega, integrating over one period of the frequency domain.[1]The DTFT converges for signals that are absolutely summable (\sum_{n=-\infty}^{\infty} |x| < \infty) or square-summable (\sum_{n=-\infty}^{\infty} |x|^2 < \infty), ensuring the frequency representation is well-defined for finite-energy discrete-time signals.[1] Key properties include linearity, time-shifting (x[n - n_0] \leftrightarrow e^{-j \omega n_0} X(e^{j\omega})), frequency-shifting (e^{j \omega_0 n} x \leftrightarrow X(e^{j(\omega - \omega_0)})), and convolution in the time domain corresponding to multiplication in the frequency domain (x * h \leftrightarrow X(e^{j\omega}) H(e^{j\omega})).[1] These properties make the DTFT essential for analyzing linear time-invariant (LTI) discrete-time systems, where the frequency response H(e^{j\omega}) characterizes system behavior.[2]Unlike the continuous-time Fourier transform, the DTFT inherently reflects the periodicity in frequency due to the discrete nature of time-domain sampling, limiting analysis to the principal range -\pi \leq \omega \leq \pi.[1] It serves as a foundational tool in digital signal processing, bridging theoretical frequency analysis with practical computations like the discrete Fourier transform (DFT), which approximates the DTFT for finite-length signals via sampling in the frequency domain.[2] Applications include filter design, spectral analysis, and understanding aliasing effects in sampled systems.[1]
Fundamentals
Definition
The discrete-time Fourier transform (DTFT) provides a frequency-domain representation of a discrete-time signal x, where n is an integer-valued time index. It is defined asX(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x e^{-j \omega n},with \omega denoting the normalized angular frequency in radians per sample.[1]This transform maps the time-domain sequence x to a continuous function X(e^{j\omega}) that is periodic with period $2\pi and represents the signal's frequency content on the unit circle in the complex plane. The magnitude |X(e^{j\omega})| indicates the amplitude of sinusoidal components at frequency \omega, while the phase \arg\{X(e^{j\omega})\} captures their relative shifts.[1][3]For the DTFT to converge in the ordinary sense to a continuous function of \omega, the signal x must satisfy the absolute summability condition \sum_{n=-\infty}^{\infty} |x| < \infty. Under this condition, the transform exists almost everywhere and is continuous. If the signal has finite energy but is not absolutely summable, the DTFT may still exist in the mean-square sense.[1][4]The DTFT emerged in the mid-20th century as a foundational tool in digital signal processing, building on earlier Fourier analysis techniques for periodic signals. Key contributions to its formalization and application were made by Alan V. Oppenheim and Ronald W. Schafer in their 1975 textbook Discrete-Time Signal Processing, which established it as a core method for analyzing discrete-time systems.[5][6]
Relation to Continuous-Time Fourier Transform
The discrete-time Fourier transform (DTFT) arises naturally from the continuous-time Fourier transform (CTFT) when analyzing sampled versions of continuous-time signals. Specifically, a discrete-time signal x can be viewed as obtained by sampling a continuous-time signal x_c(t) at uniform intervals T, yielding x = x_c(nT). This sampling process is equivalent to multiplying the continuous signal with an impulse train, forming the continuous-time signal \sum_{n=-\infty}^{\infty} x \delta(t - nT), whose CTFT is X(e^{j \Omega T}), where \Omega is the continuous-time angular frequency.[7][8]A fundamental difference between the DTFT and CTFT lies in their domains and periodicity. The CTFT operates on continuous-time signals x_c(t) and produces an aperiodic spectrum X_c(\omega) over -\infty < \omega < \infty, reflecting the infinite extent of the time domain. In contrast, the DTFT takes discrete-time inputs x and yields a spectrum X(\omega) that is continuous but 2π-periodic in the normalized frequency \omega, due to the discrete nature of the time index n. This periodicity stems from the exponential basis functions e^{j\omega n} repeating every 2π in \omega.[9][1]The DTFT's connection to the CTFT is deepened by the sampling theorem, which ensures faithful representation of bandlimited signals. If the continuous signal x_c(t) is bandlimited to frequencies below the Nyquist rate $1/(2T), its DTFT X(\omega) is a scaled, undistorted version of the CTFT X_c\left(\frac{\omega}{T}\right) within |\omega| < \pi. However, for non-bandlimited signals, sampling introduces aliasing, where spectral replicas overlap, distorting the DTFT as X(\omega) = \frac{1}{T} \sum_{k=-\infty}^{\infty} X_c\left( \frac{\omega + 2\pi k}{T} \right). This aliasing effect highlights the DTFT's role in capturing the consequences of discretization.[7][8]This relationship enables precise analysis of discrete-time systems derived from continuous ones, bypassing approximations inherent in continuous-domain modeling. By treating sampled signals directly via the DTFT, engineers can evaluate frequency responses and system behaviors in digital environments, such as filters and processors, where the periodic spectrum facilitates efficient computation without recourse to idealized continuous assumptions.[1][9]
Mathematical Formulation
Forward Transform
The forward discrete-time Fourier transform (DTFT) provides a frequency-domain representation of an aperiodic, discrete-time signal x, where n is an integer ranging from -\infty to \infty. It decomposes the signal into a continuum of complex exponentials e^{j\omega n}, with \omega denoting the normalized angular frequency in radians. This transform is particularly useful for signals that are absolutely summable, ensuring convergence of the transform sum.[1]The forward DTFT is defined by the analysis equation:X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x e^{-j\omega n},where X(e^{j\omega}) is a complex-valued function periodic with period $2\pi, emphasizing evaluation on the unit circle in the complex plane via e^{j\omega}. This formula arises from the orthogonality of the complex exponentials e^{j\omega n} over the discrete time index n. Specifically, the derivation begins with the discrete-time Fourier series (DTFS) for periodic signals, where the coefficients are given by a finite sum over one period N; as N \to \infty, the DTFS analysis equation generalizes to the infinite sum of the DTFT, with the discrete frequencies becoming continuous.[1][10]The frequency variable \omega typically spans one period from -\pi to \pi, capturing all unique frequency components due to the $2\pi-periodicity of X(e^{j\omega}). Within this range, X(e^{j\omega}) can be interpreted as the Fourier series coefficients for the periodic extension of x in the time domain, though the signal itself is aperiodic; this perspective highlights the transform's role in representing the signal's spectral content continuously. A consequence of the orthogonality underlying this decomposition is Parseval's relation, which preserves signal energy:\sum_{n=-\infty}^{\infty} |x|^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} |X(e^{j\omega})|^2 \, d\omega.This equality confirms that the total energy in the time domain equals that in the frequency domain, integrated over one period.[11]For signals that are not absolutely summable, such as periodic sequences or the unit impulse \delta, the DTFT sum may not converge in the ordinary sense. In these cases, the transform is defined in the sense of distributions, incorporating Dirac delta functions to represent impulses in the frequency domain. For instance, the DTFT of the unit impulse \delta is X(e^{j\omega}) = 1, a constant function, while for constant signals like x = 1, it involves a train of Dirac deltas: X(e^{j\omega}) = 2\pi \sum_{k=-\infty}^{\infty} \delta(\omega - 2\pi k). This distributional approach extends the DTFT to ideal or power signals while maintaining analytical utility.[12]
Inverse Transform
The inverse discrete-time Fourier transform (DTFT) recovers the original discrete-time signal x from its frequency-domain representation X(e^{j\omega}), which is a continuous, periodic function with period $2\pi. The inverse transform is given by the integralx = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(e^{j\omega}) e^{j \omega n} \, d\omegafor all integers n, where the integration is performed over any single period of X(e^{j\omega}), such as [-\pi, \pi], due to its periodicity.[1]This formula arises from the inversion of the forward DTFT through an analogy to Fourier series analysis. Consider a finite-duration sequence x extended periodically with period N, yielding a discrete-time Fourier series (DTFS) representation \tilde{X}(e^{j\omega}) = \sum_{k=0}^{N-1} \tilde{x} e^{-j \omega k}. As N \to \infty, the periodic extension in time becomes aperiodic, and the DTFS coefficients \tilde{x} approach x, while the sum over discrete frequencies transforms into the continuous integral of the DTFT, leading directly to the inverse formula.[1]If the DTFT X(e^{j\omega}) exists (e.g., for absolutely summable or square-summable sequences), the inverse uniquely determines x for all integers n, establishing a one-to-one correspondence between the time-domain signal and its frequency representation under suitable convergence conditions.[1]In practice, the continuous integral of the inverse DTFT is approximated numerically using Riemann sums over discrete frequency points \hat{\omega}_k = \frac{2\pi k}{N}, with spacing \Delta \hat{\omega} = \frac{2\pi}{N}, which yieldsx \approx \frac{1}{N} \sum_{k=0}^{N-1} X(e^{j \hat{\omega}_k}) e^{j \hat{\omega}_k n},setting the stage for computation via the discrete Fourier transform (DFT) for finite N.[13]
Properties
Algebraic and Shift Properties
The discrete-time Fourier transform (DTFT) exhibits several fundamental algebraic properties that facilitate analysis of discrete-time signals. These properties arise directly from the definition of the DTFT, given by X(\omega) = \sum_{n=-\infty}^{\infty} x e^{-j \omega n}, where \omega is the angular frequency in radians.[1]Linearity is a core property of the DTFT, stating that if x \leftrightarrow X(\omega) and y \leftrightarrow Y(\omega), then for any constants a and b,a x + b y \leftrightarrow a X(\omega) + b Y(\omega).This follows from the linearity of summation in the DTFT definition: substituting z = a x + b y yields Z(\omega) = \sum (a x + b y) e^{-j \omega n} = a \sum x e^{-j \omega n} + b \sum y e^{-j \omega n} = a X(\omega) + b Y(\omega).[1][14]The time-shifting property describes the effect of delaying or advancing a signal in time. If x \leftrightarrow X(\omega), thenx[n - n_0] \leftrightarrow e^{-j \omega n_0} X(\omega),where n_0 is an integer shift. To derive this, substitute into the DTFT: let m = n - n_0, so \sum x e^{-j \omega (m + n_0)} = e^{-j \omega n_0} \sum x e^{-j \omega m} = e^{-j \omega n_0} X(\omega). This introduces a linear phase shift proportional to the delay n_0.[1][14][15]The frequency-shifting property, also known as modulation, shifts the spectrum periodically due to the 2π-periodicity of the DTFT. If x \leftrightarrow X(\omega), thenx e^{j \omega_0 n} \leftrightarrow X(\omega - \omega_0).The proof involves direct substitution: \sum x e^{j \omega_0 n} e^{-j \omega n} = \sum x e^{-j (\omega - \omega_0) n} = X(\omega - \omega_0). This property is useful for analyzing modulated signals, with the shift wrapping around every 2π in frequency.[1][14][15]Time reversal flips the signal sequence, affecting the frequency domain symmetrically. If x \leftrightarrow X(\omega), thenx[-n] \leftrightarrow X(-\omega).Derivation: substitute m = -n, so \sum x[-m] e^{-j \omega (-m)} = \sum x e^{j \omega m}, which is X(-\omega) by replacing \omega with -\omega in the original definition (noting the even nature of the magnitude and odd phase for real signals).[1][16][14]Conjugation and scaling rules extend these properties, with conjugation stating that if x \leftrightarrow X(\omega), then x^* \leftrightarrow X^*(-\omega) for complex signals, ensuring Hermitian symmetry X^*(\omega) = X(-\omega) for real x.[1][16] A related scaling involves exponential weighting: for |\alpha| < 1, \alpha^n x \leftrightarrow X(\omega - j \ln \alpha), which hints at region-of-convergence considerations akin to the z-transform, derived by substituting \alpha^n e^{-j \omega n} = e^{-(\omega + j \ln \alpha) n} into the sum.[13][14]
Convolution and Multiplication Properties
The convolution property of the discrete-time Fourier transform (DTFT) states that the transform of the linear convolution of two discrete-time signals equals the pointwise product of their individual DTFTs. Formally, if x and y are two signals with DTFTs X(\omega) and Y(\omega), respectively, then the convolution z = (x * y) = \sum_{k=-\infty}^{\infty} x y[n - k] has DTFT Z(\omega) = X(\omega) Y(\omega). This theorem holds under the assumption that the signals are absolutely summable to ensure convergence of the DTFTs.[17][1]The proof follows by direct substitution of the convolution sum into the DTFT definition:Z(\omega) = \sum_{n=-\infty}^{\infty} z e^{-j \omega n} = \sum_{n=-\infty}^{\infty} \left( \sum_{k=-\infty}^{\infty} x y[n - k] \right) e^{-j \omega n}.Interchanging the order of summation (justified by absolute summability) yieldsZ(\omega) = \sum_{k=-\infty}^{\infty} x \left( \sum_{n=-\infty}^{\infty} y[n - k] e^{-j \omega n} \right).A change of summation variable in the inner sum, letting m = n - k, simplifies it to Y(\omega) e^{-j \omega k}, soZ(\omega) = Y(\omega) \sum_{k=-\infty}^{\infty} x e^{-j \omega k} = X(\omega) Y(\omega).This derivation highlights the utility of the DTFT in simplifying time-domain convolutions, which are computationally intensive, into frequency-domain multiplications.[17]The multiplication property provides the dual relationship: the DTFT of the pointwise product of two signals is a scaled periodic convolution of their DTFTs over one period [-\pi, \pi]. Specifically, for z = x y,Z(\omega) = \frac{1}{2\pi} \int_{-\pi}^{\pi} X(\theta) Y(\omega - \theta) \, d\theta.This integral represents the convolution wrapped periodically due to the $2\pi-periodicity of the DTFT, often arising in applications like modulation or windowing where time-domain multiplication smears the frequency content. The property can be derived analogously by expressing x and y via their inverse DTFTs and interchanging integration and summation orders.[1][16]A related correlation property involves the DTFT of the convolution of a time-reversed and conjugated signal with another: the DTFT of x[-n] * y = \sum_{k=-\infty}^{\infty} x[-(n - k)] y is X(\omega) Y^*(\omega), where ^* denotes complex conjugate. This form is particularly relevant for computing cross-correlation functions, as the cross-correlation r_{xy} = \sum_{k} x[k + n] y^* has DTFT R_{xy}(\omega) = X(\omega) Y^*(\omega). When x = y, this yields the power spectral density as |X(\omega)|^2. The derivation parallels the convolution theorem, leveraging time-reversal and conjugation properties of the DTFT.[1]These properties have significant implications for analyzing linear time-invariant (LTI) systems. The frequency response H(\omega) of an LTI system is defined as the DTFT of its impulse response h, and for an input x with output y = x * h, the output spectrum is Y(\omega) = H(\omega) X(\omega). This multiplication in the frequency domain enables straightforward computation of system responses and filter designs by shaping H(\omega) to achieve desired frequency selectivity.[1]
Signal Analysis Applications
Sampling the DTFT
The discrete-time Fourier transform (DTFT) of a sequence x is a continuous function of frequency \omega, defined over -\pi to \pi and periodic with period $2\pi. In practical signal processing, evaluating the DTFT at discrete frequencies is essential for numerical computation and analysis, leading to the discrete Fourier transform (DFT). For an N-point finite sequence, the sampling process involves assessing the DTFT X(e^{j\omega}) at equally spaced points \omega_k = 2\pi k / N for k = 0, 1, \dots, N-1. This discretization approximates the continuous frequency response with N samples, enabling finite-dimensional processing.[18][1]The N-point DFT X is thus expressed asX = \sum_{n=0}^{N-1} x e^{-j 2\pi k n / N}, \quad k = 0, 1, \dots, N-1.This formula directly corresponds to sampling the DTFT, where X \approx X(e^{j 2\pi k / N}) for sequences of length at most N. The DFT inherently assumes the input sequence is periodic with period N, representing the DTFT of the periodized version of x. Conversely, the inverse DFT recovers the finite sequence from these samples, approximating the inverse DTFT synthesis for bandlimited or finite-duration signals. This relationship allows the DFT to serve as a practical tool for frequency-domain analysis of discrete-time signals.[18][1]Finite sampling in the frequency domain imposes periodicity on the time-domain representation, which can introduce aliasing if the original sequence exceeds length N. Specifically, the periodized signal becomes \tilde{x} = \sum_{r=-\infty}^{\infty} x[n + rN], causing overlap or aliasing in time unless x = 0 for n \geq N. In the frequency domain, this manifests as the inherent $2\pi-periodicity of the DTFT being captured, but non-periodic signals with abrupt truncation at n = N lead to discontinuities. Such discontinuities result in spectral leakage, where energy from the true DTFT spreads to adjacent frequency bins due to the Gibbs phenomenon.[18][19]Truncating an infinite-duration sequence to compute the DFT is equivalent to multiplying x by a rectangular window of length N, which convolves the ideal DTFT with the window's transform—a sinc-like function with broad mainlobe and sidelobes. This convolution broadens spectral peaks and introduces ripple, exacerbating leakage for signals with sharp frequency components. To mitigate resolution limitations without altering the underlying spectrum, zero-padding extends the sequence with trailing zeros to length M > N, yielding denser DFT samples that interpolate the DTFT more finely.[20][21]Direct computation of the DFT requires O(N^2) operations, which is inefficient for large N. The fast Fourier transform (FFT) addresses this through divide-and-conquer strategies, notably the Cooley-Tukey algorithm, which decomposes the N-point DFT into smaller sub-transforms when N is composite (e.g., a power of 2), achieving O(N \log N) complexity. This enables efficient evaluation of DTFT samples for real-time applications like filtering and spectral estimation.[22]
Periodic Signals
In discrete-time signal processing, periodic signals play a fundamental role in understanding the behavior of the discrete-time Fourier transform (DTFT). A discrete-time signal x is periodic with period N if x[n + N] = x for all integers n, where N is a positive integer. The DTFT of such a signal reveals a spectrum composed entirely of discrete frequency components, reflecting the inherent repetition in the time domain.[1]The DTFT of a periodic signal x with period N is given byX(\omega) = 2\pi \sum_{k=-\infty}^{\infty} c_k \delta\left( \omega - \frac{2\pi k}{N} \right),where c_k are the Fourier series coefficients of x, defined as c_k = \frac{1}{N} \sum_{n=0}^{N-1} x e^{-j 2\pi k n / N}. This expression indicates that the spectrum consists of an infinite train of impulses located at harmonically related frequencies \omega_k = 2\pi k / N, with strengths proportional to the coefficients c_k. The periodicity of the DTFT with period $2\pi ensures that the impulses repeat every $2\pi, but for the periodic signal, the non-zero impulses are confined to multiples of $2\pi / N.[1]This impulse train representation arises because a periodic signal can be expressed as a sum of complex exponentials at the fundamental frequency and its harmonics: x = \sum_{k=-\infty}^{\infty} c_k e^{j 2\pi k n / N}. The DTFT of each exponential e^{j \omega_0 n} is $2\pi \sum_{m=-\infty}^{\infty} \delta(\omega - \omega_0 - 2\pi m), leading to the overall line spectrum when superposed. Such spectra are characteristic of periodic discrete-time signals, contrasting with the continuous spectra of aperiodic signals.[1][23]The inverse DTFT recovers the original periodic signal throughx = \sum_{k=-\infty}^{\infty} c_k e^{j 2\pi k n / N},which follows from integrating the impulse train over one period [-\pi, \pi], effectively summing the contributions from the impulses within that interval due to the $2\pi-periodicity. This synthesis equation directly links the DTFT to the discrete-time Fourier series, emphasizing that the transform encapsulates the harmonic decomposition.[1]For aperiodic signals, the DTFT can be viewed as the limiting case of the periodic representation as the period N \to \infty. In this limit, the discrete spectral lines become infinitely dense, transforming the impulse train into a continuous function, and the Fourier series sum evolves into the inverse DTFT integral. This perspective unifies the treatment of periodic and aperiodic discrete-time signals under the DTFT framework.[1]In applications, the DTFT of periodic signals facilitates the analysis of cyclic processes in digital signal processing, such as in the study of repeating patterns in sampled data from rotating machinery or periodic noise sources. It is particularly useful for tone generation, where synthesizing sinusoids at specific harmonics via the inverse transform enables the creation of periodic audio signals or test tones in communication systems.[13]
Transform Relationships
Connection to Z-Transform
The z-transform of a discrete-time signal x is defined asX(z) = \sum_{n=-\infty}^{\infty} x z^{-n},where z is a complex variable and the sum converges within a region of convergence (ROC) in the complex z-plane.[24] The discrete-time Fourier transform (DTFT) is obtained by evaluating the z-transform on the unit circle, i.e., X(e^{j\omega}) = X(z)|_{z = e^{j\omega}}, provided that the unit circle |z| = 1 lies within the ROC.[25]The DTFT exists only if the ROC of the z-transform includes the unit circle; otherwise, the frequency response may not converge in the ordinary sense.[25] Poles and zeros of X(z) in the z-plane determine the behavior of the frequency response X(e^{j\omega}) along the unit circle, with poles near the unit circle causing sharp resonances in the magnitude response.[26]The z-transform offers advantages over the DTFT by representing signals as Laurent series in the complex plane, allowing analysis of sequences where the DTFT does not converge due to insufficient decay.[27] It facilitates the study of rational transfer functions H(z) = \frac{\sum_{k=0}^{M} b_k z^{-k}}{1 + \sum_{k=1}^{N} a_k z^{-k}}, which model linear time-invariant systems and enable pole-zero analysis for filter design.[26]Conversely, the z-transform generalizes the DTFT by extending the evaluation from the unit circle to the entire complex plane, permitting stabilityanalysis through pole locations: a system is bounded-input bounded-output stable if all poles lie inside the unit circle and the ROC includes |z| = 1.[25] For causal signals, where x = 0 for n < 0, the DTFT can be derived from the one-sided z-transform X(z) = \sum_{n=0}^{\infty} x z^{-n} by restricting to the right-half plane and evaluating on the unit circle when the ROC permits.[28]
Connection to Discrete Fourier Transform
The discrete Fourier transform (DFT) of a finite-length sequence x of length N is equivalent to equally spaced samples of the discrete-time Fourier transform (DTFT) of the infinite sequence obtained by periodically extending x with period N. This periodic extension implies that the DFT inherently assumes the underlying signal is periodic, which introduces time-domain aliasing if the original finite sequence is aperiodic, as the tails of adjacent periods overlap.[18]As the length N approaches infinity for an aperiodic signal, the DFT samples become denser and converge to the continuous DTFT, providing a theoretical bridge between finite numerical computation and the ideal frequency representation. Zero-padding the finite sequence—increasing N by appending zeros—mitigates time-domain aliasing by extending the period without altering the signal content, thereby reducing overlap in the periodic replicas and yielding a finer sampling of the DTFT that better approximates its continuous nature.[29][30]Both the DTFT and DFT preserve signal energy via Parseval's theorem, but with differing normalizations: for the DTFT, \sum_{n=-\infty}^{\infty} |x|^2 = \frac{1}{2\pi} \int_{-\pi}^{\pi} |X(e^{j\omega})|^2 d\omega, whereas for the DFT, \sum_{n=0}^{N-1} |x|^2 = \frac{1}{N} \sum_{k=0}^{N-1} |X|^2, reflecting the finite discrete framework. In practice, the DTFT serves theoretical analysis of infinite or aperiodic signals, such as deriving filter properties, while the DFT enables numerical implementation for applications like digital filtering and spectrogram computation on finite data.[18]Approximating the DTFT via a truncated DFT, such as through windowing or finite summation, can exhibit the Gibbs phenomenon near discontinuities in the frequency response, where overshoots and ripples persist regardless of increasing N, with an amplitude of approximately 9% of the jump height. This error arises from the incomplete cancellation of high-frequency components in the periodic extension and is particularly evident in ideal filter approximations.[31]
Common Transforms
Table of Transforms
The following table summarizes selected common discrete-time Fourier transform (DTFT) pairs, where the DTFT is defined as X(\omega) = \sum_{n=-\infty}^{\infty} x e^{-j \omega n}. These pairs are drawn from standard references in discrete-time signal processing.[1][16]
To illustrate the computation of the discrete-time Fourier transform (DTFT), consider the finite rectangular pulse defined as x = 1 for |n| \leq M and x = 0 otherwise, where M is a non-negative integer. This signal has a duration of $2M + 1 samples. The DTFT is given by the forward transform formula:X(\omega) = \sum_{n=-\infty}^{\infty} x e^{-j \omega n} = \sum_{n=-M}^{M} e^{-j \omega n}.[1]This sum is a finite geometric series with first term e^{j \omega M}, common ratio e^{-j \omega}, and $2M + 1 terms. The sum of such a series is \frac{\sin(((2M+1)\omega)/2)}{\sin(\omega/2)}, derived by multiplying the series by the common ratio and subtracting, then solving for the original sum, followed by applying the identity for the sine of sums. Thus,X(\omega) = \frac{\sin(((2M+1)\omega)/2)}{\sin(\omega/2)}.This expression is periodic with period $2\pi and represents a discrete-time analog of the sinc function.[1]Next, examine the causal exponential decay signal x = a^n u, where |a| < 1 ensures convergence and u is the unit step function (u = 1 for n \geq 0, 0 otherwise). The DTFT computation yieldsX(\omega) = \sum_{n=0}^{\infty} a^n e^{-j \omega n} = \sum_{n=0}^{\infty} (a e^{-j \omega})^n.This is an infinite geometric series with ratio r = a e^{-j \omega} and |r| < 1, summing to \frac{1}{1 - a e^{-j \omega}}.[1]To demonstrate the time-shift property, consider a delayed version x_d = x[n - n_0] = a^{n - n_0} u[n - n_0] for integer n_0 > 0. The DTFT of a shifted signal is X_d(\omega) = e^{-j \omega n_0} X(\omega), soX_d(\omega) = \frac{e^{-j \omega n_0}}{1 - a e^{-j \omega}}.This phase shift e^{-j \omega n_0} linearly varies with frequency, reflecting the delay.[1]For an example involving convolution, take the convolution of two identical rectangular pulses, each of length L = 2M + 1 centered at the origin: y = (x * x) = \sum_{k} x x[n - k]. This results in a triangular pulse of length $2L - 1, with y rising linearly from 0 to L at n = 0 and then falling symmetrically. By the convolution theorem for DTFT, Y(\omega) = X(\omega) \cdot X(\omega) = \left( \frac{\sin(((2M+1)\omega)/2)}{\sin(\omega/2)} \right)^2, verifying that the transform is the squared magnitude of the individual sinc-like functions, producing a narrower main lobe and faster side-lobe decay.[1]The frequency response of these transforms provides intuition through their magnitude and phase. For the rectangular pulse, the magnitude |X(\omega)| exhibits a periodic sinc-like envelope with the main lobe width approximately \frac{4\pi}{2M+1} (inversely proportional to duration), zeros at multiples of \frac{2\pi}{2M+1}, and sidelobes decaying as $1/|\omega|; the phase is odd and linear within the main lobe, indicating symmetry. For the exponential decay, |X(\omega)| = \frac{1}{|1 - a e^{-j \omega}|} is low-pass with maximum at \omega = 0 and roll-off depending on |a| (closer to 1 yields slower decay), while the phase is the argument of the denominator, showing a nonlinear, frequency-dependent shift.[1]