Continuous- and Discrete-Time Fourier Transforms

Introduction and context

Fourier analysis represents a signal as a weighted combination of complex sinusoids. In signals and systems, this change from the time domain to the frequency domain is fundamental because differentiation, convolution, modulation, filtering, and spectral analysis often become simpler after transformation. The continuous-time Fourier transform (CTFT) is used for continuous-time signals, whereas the discrete-time Fourier transform (DTFT) is used for sequences.

The transforms are not merely computational devices. They provide a language for describing bandwidth, resonance, spectral occupancy, interference, noise, sampling, and filter selectivity. At postgraduate level, it is essential to understand not only transform pairs but also convergence, generalized functions, periodicity, symmetry, numerical implementation, and the distinction between the DTFT and the discrete Fourier transform (DFT).

Learning objectives

After studying this chapter, the reader should be able to:

1. Define and interpret the CTFT and DTFT.
2. Derive and use important transform pairs and properties.
3. Explain convergence, energy, power, and generalized spectra.
4. Analyze linear time-invariant systems in the frequency domain.
5. Relate sampling to spectral replication and aliasing.
6. Distinguish the DTFT, DFT, and fast Fourier transform (FFT).
7. Apply Fourier methods to practical communication and signal-processing problems.
8. Evaluate assumptions, limitations, and implementation issues.

1. The Fourier representation

A complex exponential e^jΩt is an eigenfunction of every linear time-invariant continuous-time system. If the input is e^jΩt, the output is the same exponential multiplied by a complex number H(jΩ). The magnitude of this number determines amplitude scaling and its phase determines phase shift.

For a continuous-time signal x(t), the CTFT convention used in this chapter is

and the inverse transform is

Here, t is time in seconds, Ω is angular frequency in radians per second, and j = √(−1). The transform variable X(jΩ) generally has units equal to the units of x(t) multiplied by seconds.

For a discrete-time sequence x[n], where n is an integer, the DTFT is

with inverse

The variable ω is angular frequency in radians per sample. Unlike the CTFT, the DTFT is periodic:

This periodicity follows because e^{−j(ω+2π)n} = e^−jωn for integer n.

2. Existence and convergence

The simplest sufficient condition for the CTFT to exist as an ordinary function is absolute integrability:

For the DTFT, the corresponding condition is

These are sufficient, not necessary, conditions. Signals such as sinusoids and constants are not absolutely integrable, yet they possess Fourier transforms in the sense of distributions. For example, under the stated CTFT convention,

where δ(Ω) is the Dirac delta distribution. A continuous-time complex sinusoid has transform

A discrete-time sinusoid has a periodic line spectrum. For x[n] = e^jω₀n,

The delta notation must be interpreted distributionally; it is not an ordinary finite-valued function.

Energy and power provide useful convergence classifications. Continuous-time energy is

and discrete-time energy is

A signal is an energy signal if its energy is finite and nonzero. A power signal has finite nonzero average power but usually infinite total energy. For a continuous-time signal, average power is

For a discrete-time signal, it is

3. Important CTFT pairs

Using u(t) for the unit step and rect(t/T) for a rectangular pulse of width T, important pairs include:

Here, sinc(z)=sin(z)/z, with sinc(0)=1. The rectangular-pulse pair assumes the pulse equals one for |t|<T/2 and zero elsewhere, apart from endpoint conventions.

For a decaying exponential,

has transform

The condition a>0 ensures decay and convergence. Its magnitude and phase are

4. Important DTFT pairs

For x[n]=aⁿ u[n], the transform follows from a geometric series:

The restriction |a|<1 is necessary for ordinary convergence. If a is real and positive, the spectrum is largest near ω=0.

5. Fundamental transform properties

Let x(t) ⇌ X(jΩ) and y(t) ⇌ Y(jΩ). The principal properties are:

Linearity

Time shift

A time shift changes phase but not magnitude.

Frequency shift (modulation)

Multiplication by a sinusoid creates translated spectral copies. For cosine modulation,

Time scaling

Compression in time expands the spectrum and reduces its spectral amplitude by the factor 1/|a|.

Differentiation

This assumes boundary terms vanish or that the relation is interpreted distributionally.

Convolution

where (x*h)(t)=∫x(τ)h(t−τ)dτ. Conversely,

The corresponding DTFT convolution property is

where convolution is a sum over integer shifts.

Parseval’s relation

For finite-energy continuous-time signals,

For discrete-time signals,

These relations state that energy is preserved under the transform, subject to the normalization convention.

6. LTI systems and frequency response

For an LTI system with impulse response h(t),

and therefore

The frequency response is

If the system is stable in the bounded-input, bounded-output sense, a standard sufficient condition is ∫|h(t)|dt<∞. For a discrete-time system, the corresponding condition is ∑|h[n]|<∞.

The magnitude response |H(jΩ)| describes amplitude alteration, while ∠H(jΩ) describes phase alteration. Group delay is defined, where differentiable, by

It measures the frequency-dependent delay of a narrowband component. Constant group delay is desirable when waveform shape must be preserved.

Worked example: first-order low-pass system

Let

Then

At Ω=0, the gain is 1/a. The half-power angular frequency satisfies

Since |H(jΩ)|²=1/(a²+Ω²), this gives Ω_c=a. Thus a is the 3 dB bandwidth in radians per second for this unnormalized first-order response.

7. Symmetry and spectral interpretation

If x(t) is real-valued, then

This is conjugate symmetry. Consequently, the magnitude spectrum is even and the phase spectrum is odd, apart from phase discontinuities. If x(t) is real and even, its transform is real and even. If it is real and odd, its transform is imaginary and odd.

For discrete-time real sequences, the same principle applies over one period:

These symmetries reduce computation and provide valuable error checks in analytical and numerical work.

8. Sampling and spectral replication

Sampling a continuous-time signal at rate f_s samples per second, with sampling interval T_s=1/f_s, can be modeled as multiplication by an impulse train:

The impulse train has Fourier transform

where Ω_s=2π/T_s. Therefore,

Sampling replicates the original spectrum at integer multiples of the sampling angular frequency. If the replicas overlap, aliasing occurs and cannot generally be removed after sampling.

For a bandlimited signal with highest frequency f_max, ideal sampling requires

This is the Nyquist sampling condition. In practical systems, an analog anti-aliasing low-pass filter is applied before the analog-to-digital converter, because real signals are rarely perfectly bandlimited and real filters are not ideal.

9. DTFT, DFT, and FFT

The DTFT is a continuous function of ω and is periodic. The DFT of an N-point sequence is a finite set of samples:

The inverse DFT is

The DFT assumes that the analyzed block is one period of a periodic sequence. Consequently, truncating a nonperiodic signal introduces spectral leakage. Windowing reduces sidelobes but broadens the main lobe; it is therefore a trade-off between leakage suppression and frequency resolution.

The FFT is not a different transform. It is a family of algorithms for computing the DFT efficiently. Direct DFT evaluation requires order N² complex operations, whereas radix-2 FFT methods require order N log₂N operations when N is a power of two.

Zero-padding interpolates the sampled display of a spectrum but does not increase the information or the fundamental resolution determined by the observation duration. Frequency-bin spacing is

where T_obs=N/f_s is the observation duration for an unwindowed block.

10. Worked numerical example: DTFT of a finite sequence

Consider x[n]=1 for n=0,1,2,3 and zero otherwise. Then

Using the finite geometric-series identity,

where the value at points with a zero denominator is obtained by continuity. At ω=0, the original sum gives X(e^j0)=4. The magnitude is therefore largest at zero frequency. The factor e^−j3ω/2 represents linear phase associated with the sequence’s midpoint at n=1.5.

11. Applied case: vibration monitoring of a rotating machine

Background. Consider an illustrative industrial case involving a motor-driven pump. An accelerometer produces a voltage proportional to casing acceleration. The maintenance team wants to detect imbalance and bearing degradation before mechanical failure.

Problem. The motor rotates at 1,800 revolutions per minute, corresponding to

A characteristic bearing fault is expected near 240 Hz, with harmonics possible. The acquisition system samples at 1 kHz.

Analysis. The Nyquist frequency is 500 Hz, so the 240 Hz component is theoretically observable. However, sensor noise and structural resonances may exist above 500 Hz. An analog low-pass anti-aliasing filter is therefore required before conversion. A record of 2,000 samples has duration

and frequency-bin spacing

This resolution is adequate to distinguish the 30 Hz rotational component from nearby components. A Hann window may be selected if the record does not contain an integer number of rotational cycles. The FFT magnitude is calibrated according to the sensor sensitivity, sampling rate, window coherent gain, and scaling convention.

Approach and solution. The engineering workflow is: calibrate the sensor; remove the mean if appropriate; apply the anti-aliasing filter; acquire synchronized blocks; select a window; compute the FFT; inspect the 30 Hz fundamental, harmonics, and the 240 Hz fault band; and compare trends against a baseline. Repeated measurements under comparable load are more informative than a single spectrum.

Outcome. A rising narrowband component near 240 Hz, especially if accompanied by harmonics and sidebands, can justify inspection. Fourier evidence alone does not prove a particular mechanical fault; it must be combined with operating conditions, time-domain impulsiveness, phase information, and physical inspection.

Learning points. The case demonstrates spectral interpretation, sampling constraints, windowing, resolution, calibration, and the importance of combining signal evidence with domain knowledge. Automated or AI-based anomaly detection can rank abnormal spectra, but it should preserve raw data, expose confidence measures, and remain subject to engineering validation rather than replacing safety-critical judgment.

12. Limitations, implementation, and best practice

Fourier transforms assume a useful representation in stationary complex sinusoids. Rapidly changing events may be poorly localized in frequency. Short-time Fourier analysis, wavelets, or time-frequency distributions may then be preferable. Spectral leakage, aliasing, quantization noise, finite precision, sensor saturation, and incorrect normalization can produce misleading results.

Best practice includes:

• State the transform convention and units.
• Check convergence or explicitly use distribution theory.
• Apply anti-aliasing before sampling.
• Record sampling rate, duration, window, overlap, and scaling.
• Distinguish amplitude, power, and power spectral density.
• Use symmetry and Parseval checks to validate calculations.
• Avoid interpreting zero-padding as increased physical resolution.
• Compare spectra under equivalent operating conditions.
• Preserve metadata and raw data for reproducibility.
• Treat automated classification as decision support, especially in safety-critical systems.

In communications, Fourier analysis supports modulation, channel equalization, orthogonal frequency-division multiplexing, spectrum regulation, and interference analysis. In biomedical engineering it supports ECG, EEG, and imaging analysis, although nonstationarity and patient privacy require particular care. In wireless systems, spectral occupancy must also comply with regulatory emission masks and coexistence requirements.

Chapter Summary

The CTFT represents continuous-time signals as continuous-frequency components, while the DTFT represents discrete-time sequences with a spectrum periodic in 2π. Transform existence depends on convergence conditions, and sinusoids or constants may require generalized functions. Linearity, shifting, scaling, modulation, differentiation, convolution, and Parseval’s relation form the main analytical toolkit.

For LTI systems, convolution in time becomes multiplication in frequency, making H(jΩ) the central frequency-response function. Sampling produces periodic spectral replicas, and overlap causes irreversible aliasing. The DFT samples a finite, periodic-spectrum representation, while the FFT is an efficient algorithm for computing it. Finite observation, windowing, leakage, and resolution must be considered in practical spectral measurements. Reliable engineering use requires correct units, calibration, anti-aliasing, reproducible processing, and critical interpretation.

Key Terms

CTFT: Fourier transform of a continuous-time signal.

DTFT: Fourier transform of a discrete-time sequence.

Angular frequency: Frequency measured in radians per second or radians per sample.

Dirac delta: Generalized function representing an ideal impulse.

Convolution: Integral or sum describing the output of an LTI system.

Frequency response: Transform of an LTI system’s impulse response.

Parseval’s relation: Equality relating signal energy in time and frequency domains.

Conjugate symmetry: Spectral symmetry produced by a real-valued signal.

Modulation: Multiplication by an oscillatory signal that shifts spectral content.

Sampling: Conversion from continuous time to discrete-time measurements.

Aliasing: Irreversible spectral overlap caused by insufficient sampling rate.

Nyquist rate: Twice the highest frequency of a bandlimited signal.

DFT: Finite set of Fourier coefficients for a finite data block.

FFT: Efficient algorithm for computing the DFT.

Spectral leakage: Spreading caused by finite observation or nonperiodic block boundaries.

Window: Weighting function applied to a finite data record.

Frequency resolution: Ability to distinguish nearby spectral components.

Group delay: Negative derivative of phase with respect to angular frequency.

Power spectral density: Distribution of signal power over frequency.

Anti-aliasing filter: Pre-sampling filter that attenuates frequencies above the usable band.

Review Questions

Short Answer Questions

1. Why is the DTFT periodic in frequency with period 2π?
2. State the CTFT convolution property and explain its systems significance.
3. What is the difference between signal energy and average power?
4. Define spectral leakage and identify one method for reducing it.
5. Why is an anti-aliasing filter used before an analog-to-digital converter?

Long Answer Questions

1. Derive the CTFT of e^−atu(t) for a>0, and interpret its magnitude and phase.
2. Explain the sampling theorem using spectral replication and discuss the consequences of aliasing.
3. Compare the CTFT, DTFT, DFT, and FFT in terms of signal type, frequency variable, periodicity, and computation.
4. Derive the frequency response of a first-order continuous-time LTI system and explain its bandwidth.
5. Discuss windowing, zero-padding, frequency resolution, and leakage in practical FFT-based measurement.

Analytical / Application Questions

1. A signal is sampled at 8 kHz and contains components at 1.2 kHz and 5.1 kHz. Determine which component aliases into the first Nyquist interval and calculate its apparent frequency.
2. An FFT uses 4,096 samples at a sampling rate of 20 kHz. Calculate the observation duration and bin spacing, then explain whether zero-padding changes the physical resolution.
3. A real sensor signal has a strong line at 60 Hz and symmetric spectral components in a two-sided plot. Explain how conjugate symmetry can be used to validate the measurement and identify two possible causes of unexpected asymmetry.

Multiple Choice Questions with Answers

Question 1

Which statement correctly describes the DTFT?

A. It is periodic in frequency with period 2π

B. It is defined only at N discrete frequency points

C. It exists only for periodic sequences

D. It always has finite duration

Correct Answer: A. It is periodic in frequency with period 2π

Question 2

Under the CTFT convention in this chapter, the transform of δ(t) is

A. 2πδ(Ω)

B. 1

C. jΩ

D. 1/(jΩ)

Correct Answer: B. 1

Question 3

For x(t−t₀), the CTFT is

A. X(j(Ω−t₀))

B. e^{jΩt₀}X(jΩ)

C. e^{−jΩt₀}X(jΩ)

D. X(jΩ−t₀)

Correct Answer: C. e^{−jΩt₀}X(jΩ)

Question 4

If X(jΩ) is the CTFT of x(t), the CTFT of dx(t)/dt is

A. X(jΩ)/(jΩ)

B. jΩX(jΩ)

C. X(jΩ+j)

D. Ω²X(jΩ)

Correct Answer: B. jΩX(jΩ)

Question 5

For an LTI system, time-domain convolution corresponds in frequency to

A. Addition

B. Differentiation

C. Multiplication

D. Time reversal

Correct Answer: C. Multiplication

Question 6

A sinusoid at 3 kHz is sampled at 4 kHz. Its apparent frequency in the first Nyquist interval is

A. 0.5 kHz

B. 1 kHz

C. 3 kHz

D. 7 kHz

Correct Answer: B. 1 kHz

Question 7

The frequency-bin spacing for 2,000 samples acquired at 1 kHz is

A. 0.2 Hz

B. 0.5 Hz

C. 2 Hz

D. 500 Hz

Correct Answer: B. 0.5 Hz

Question 8

Which statement about zero-padding is correct?

A. It increases the original observation duration

B. It eliminates aliasing

C. It interpolates the displayed DFT spectrum without adding new measured information

D. It always narrows the true main lobe

Correct Answer: C. It interpolates the displayed DFT spectrum without adding new measured information

Question 9

For a real-valued continuous-time signal, the spectrum generally satisfies

A. X(−jΩ)=X(jΩ)

B. X(−jΩ)=−X(jΩ)

C. X(−jΩ)=X*(jΩ)

D. X(−jΩ)=1/X(jΩ)

Correct Answer: C. X(−jΩ)=X*(jΩ)

Question 10

The radix-2 FFT reduces typical computational order from approximately

A. N to log N

B. N² to N log₂N

C. log N to N²

D. N log₂N to N³

Correct Answer: B. N² to N log₂N