DFT, FFT, and Computational Spectral Analysis

Introduction and Context

The discrete Fourier transform (DFT) is one of the central computational tools in digital signal processing. It converts a finite sequence of time-domain samples into a finite set of frequency-domain coefficients. This transformation enables engineers to identify spectral components, measure bandwidth, design filters, estimate system responses, detect periodicity, and implement convolution efficiently.

The DFT is mathematically derived from the discrete-time Fourier transform (DTFT), but it is not identical to it. The DTFT is generally a continuous function of frequency, whereas the DFT consists of a finite number of samples of a periodic frequency-domain representation. The fast Fourier transform (FFT) is not a separate transform; it is a family of efficient algorithms for computing the DFT.

At postgraduate level, it is important to understand not only the formulas but also the assumptions behind spectral interpretation. Sampling, finite observation time, periodic extension, leakage, windowing, aliasing, zero-padding, numerical precision, and computational complexity all influence the reliability of a spectrum.

Learning Objectives

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

1. Derive and interpret the DFT and inverse DFT.
2. Relate sampling frequency, record length, frequency-bin spacing, and observation duration.
3. Explain periodic extension, spectral leakage, scalloping loss, and window effects.
4. Distinguish the DFT from the DTFT, FFT, and related spectral estimators.
5. Derive the radix-2 decimation-in-time FFT structure.
6. Use DFT properties for circular convolution, correlation, and system analysis.
7. Construct appropriately scaled amplitude, power, and one-sided spectra.
8. Evaluate practical spectral-analysis errors and limitations.
9. Apply computational spectral analysis to an engineering problem.

1. From the DTFT to the DFT

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

where n is an integer sample index and ω is angular frequency in radians per sample. The DTFT is periodic with period 2π because e^{-j(ω+2π)n} = e^{-jωn} for integer n.

In practical measurement, only N samples are usually available. Let the observed sequence be x[0], x[1], …, x[N−1]. The N-point DFT is defined by

The inverse DFT (IDFT) is

Here, x[n] is the finite input sequence, X[k] is the kth complex spectral coefficient, N is the transform length, and j = √−1. The normalization convention used above places the factor 1/N in the inverse transform. Other conventions distribute scaling differently, but the forward and inverse definitions must remain mutually consistent.

The DFT can be interpreted in two equivalent ways:

• It samples the DTFT of a finite-duration sequence at frequencies ω_k = 2πk/N.
• It computes the Fourier-series coefficients of the periodic extension of the N-sample record.

The second interpretation is especially important. The DFT assumes that the observed record repeats periodically. If the first and last samples do not join smoothly, the periodic extension contains an artificial discontinuity. That discontinuity produces energy over many frequency bins, known as spectral leakage.

2. Frequency Mapping and Resolution

Suppose the sampling frequency is f_s samples per second. The DFT frequency associated with bin k is

The frequency spacing is

where T = N/f_s is the observation duration. Thus, frequency spacing is controlled by both the sampling rate and the duration of observation. Increasing N by increasing the observation duration improves bin spacing. Increasing N by zero-padding does not improve the fundamental ability to resolve two nearby tones; it only produces a denser display of the underlying sampled spectrum.

For a real-valued time sequence, the DFT has conjugate symmetry:

where * denotes complex conjugation. Consequently, positive- and negative-frequency components contain redundant information. For even N, a one-sided spectrum normally includes bins 0 through N/2, corresponding to frequencies from 0 to f_s/2.

The bin k = 0 is the DC component. For even N, k = N/2 represents the Nyquist frequency, f_s/2. Frequencies above f_s/2 in the unshifted DFT are commonly interpreted as negative frequencies: bin k corresponds to f_k − f_s when k > N/2.

Resolution versus sampling rate

Sampling frequency determines the highest uniquely representable frequency, while record duration determines nominal frequency spacing. For example, if f_s = 10 kHz and N = 2000, then T = 0.2 s and Δf = 5 Hz. The Nyquist frequency is 5 kHz. If the same N samples are acquired at 20 kHz, Δf becomes 10 Hz and the Nyquist frequency becomes 10 kHz; the frequency grid is wider, not finer.

3. DFT as a Matrix Transformation

The DFT may be represented as a matrix operation. Define the primitive Nth root of unity as

Then

or, in vector form,

where F_N is the N × N DFT matrix. Its entries are [F_N]_{k,n} = W_N^{kn}. The inverse relation is

where H denotes the conjugate transpose. The orthogonality of complex exponentials gives

and 0 when k and m are distinct modulo N. This orthogonality explains why a sequence consisting exactly of a DFT-bin sinusoid appears in only the corresponding spectral bins, subject to the symmetry required for real signals.

A unitary DFT uses 1/√N in both the forward and inverse transforms. This convention preserves the Euclidean norm directly, but engineering software often uses the forward-unscaled, inverse-1/N convention.

4. Important DFT Properties

Let x[n] ↔ X[k] and y[n] ↔ Y[k] denote an N-point DFT pair.

Linearity

where a and b may be real or complex constants.

Circular time shift

If y[n] = x[(n−n_0) mod N], then

A circular shift changes phase but not magnitude.

Circular frequency shift

Multiplication by a complex exponential in time shifts the DFT circularly in frequency:

Parseval’s relation

For the normalization used in this chapter,

If x[n] has units of volts, the left side has units of volt². The frequency-domain sum has the same units because the factor 1/N is dimensionless. Energy and power interpretations require care when the sampling interval is included.

Circular convolution

The N-point circular convolution of x[n] and h[n] is

and satisfies

This property is computationally useful, but it differs from ordinary linear convolution. To obtain linear convolution using DFTs, both sequences must be zero-padded to a length at least L_x + L_h − 1, where L_x and L_h are their lengths.

5. Spectral Leakage and Windowing

Consider a sampled sinusoid x[n] = A cos(2πf_0n/f_s + φ). If f_0 is exactly an integer multiple of Δf, its positive-frequency component aligns with a DFT bin. If it is not, the finite record truncates the sinusoid and the DFT energy spreads among nearby bins. This is spectral leakage.

The observed record can be written as the product of an underlying sequence and a window w[n]:

Multiplication in time corresponds to periodic convolution in frequency. Thus, the spectrum of the window controls the shape of leakage around a tone.

The rectangular window, w[n] = 1 for 0 ≤ n ≤ N−1, has the narrowest main lobe among many common windows but relatively large sidelobes. Windows such as Hann, Hamming, Blackman, and flat-top reduce sidelobes or improve amplitude accuracy at the cost of a wider main lobe.

Window choice is an engineering trade-off. If detecting a weak tone near a strong tone is the priority, sidelobe suppression is important. If separating closely spaced tones is the priority, a narrower main lobe may be preferable.

Coherent gain

A window reduces the measured amplitude of a bin-centered sinusoid. Its coherent gain is

For a real sinusoid of peak amplitude A that is exactly bin-centered, an appropriately scaled two-sided DFT magnitude is approximately A G_c/2 at each positive and negative frequency line. Correct amplitude estimation therefore requires compensation for G_c, along with the usual one-sided factor of two where applicable.

6. Scaling Spectra Correctly

A raw DFT magnitude is not automatically an amplitude spectrum, power spectrum, or power spectral density. The desired physical interpretation determines the scaling.

For an unwindowed, real-valued sinusoid of peak amplitude A that lies exactly at a non-DC, non-Nyquist bin, the two-sided DFT contains magnitude approximately AN/2 at each spectral line. A common one-sided peak-amplitude estimate is therefore

for 1 ≤ k < N/2, assuming no window and coherent sampling. The factor of two combines positive and negative frequency energy. It must not be applied to the DC or Nyquist bins in an even-length transform.

For a windowed record, a basic coherent-gain correction is

under the same bin-centered assumptions. For noise and power measurements, equivalent noise bandwidth (ENBW), not only coherent gain, is relevant. In normalized frequency-bin units,

The physical ENBW is ENBW × Δf. A power spectral density estimate must account for sampling frequency, window power, and the chosen normalization. Software documentation should always be checked before comparing spectra produced by different packages.

7. The Fast Fourier Transform

Direct DFT evaluation requires approximately N² complex multiplications and additions. The FFT exploits symmetry and periodicity in the factors W_N^{kn} to reduce the computational cost to order N log₂N for radix-2 transforms.

Radix-2 decimation-in-time derivation

Assume N is even. Separate the DFT sum into even and odd indices:

Since W_N^{2k} = W_{N/2}^k,

where

and

are DFTs of the even- and odd-indexed subsequences. Using periodicity,

for 0 ≤ k < N/2. Thus one N-point DFT is reduced to two N/2-point DFTs and N/2 combination operations. Repeating recursively produces the radix-2 FFT.

The basic computation is called a butterfly:

For N = 2^m, there are m stages, each containing N/2 butterflies. The number of stages is log₂N, explaining the order N log₂N complexity.

Decimation-in-frequency and other FFTs

Decimation-in-frequency splits the output frequency indices rather than the input time indices. Radix-4 and mixed-radix algorithms are efficient when N has factors other than two. Bluestein’s algorithm handles arbitrary transform lengths by converting a DFT into a convolution, while prime-factor and split-radix methods exploit particular factorizations. Modern libraries often select algorithms automatically.

FFT speed also depends on memory access, cache behavior, vectorization, parallel processing, hardware accelerators, and numerical precision. An asymptotically efficient algorithm may not be fastest for every transform length.

8. Computational Spectral-Analysis Workflow

A defensible spectral analysis normally follows these steps:

1. Define the physical question: tone detection, bandwidth estimation, fault diagnosis, noise characterization, or system identification.
2. Select the sampling frequency. Ensure that the anti-aliasing filter and Nyquist requirement are satisfied.
3. Remove offsets or trends when appropriate. A large mean can dominate the DC bin and obscure low-frequency components.
4. Select record length from the required observation duration and frequency separation.
5. Choose a window based on the resolution-versus-leakage trade-off.
6. Compute the DFT, usually through an FFT.
7. Apply correct amplitude, power, or PSD scaling.
8. Inspect positive and negative frequencies consistently.
9. Validate results using known tones, energy checks, repeated records, or an alternative method.
10. Report sampling rate, record length, window, overlap, scaling, averaging method, and units.

Zero-padding

If an N-sample sequence is extended with zeros to length M > N, the DFT frequency spacing becomes f_s/M, but the observation duration remains N/f_s. Zero-padding interpolates the periodic spectrum on a finer frequency grid; it does not create new information or improve true resolution. It can, however, improve graphical peak localization and facilitate interpolation-based frequency estimation.

Peak interpolation

When a tone lies between bins, the largest DFT bin is not generally the true frequency. Interpolation using neighboring magnitudes can improve estimates. Such methods depend on the window and on assumptions about isolated tones and noise. They should not be interpreted as replacing a longer observation record.

Short-time Fourier analysis

For nonstationary signals, a single DFT averages behavior over an entire record. The short-time Fourier transform (STFT) applies a window that moves through the signal:

where m is the frame index, R is the hop size in samples, and w[n] is the analysis window. Smaller windows improve time localization but worsen frequency resolution. Overlap is commonly used to reduce gaps between frames, especially with windows satisfying suitable constant-overlap-add conditions.

9. Worked Example: Estimating a Sinusoidal Component

Assume a measurement system samples a signal at f_s = 8 kHz for N = 1024 samples. The record duration is

The bin spacing is

Suppose the signal contains a 1 kHz sinusoid of peak amplitude 0.8 V, and the frequency is exactly a DFT-bin frequency. Since

the tone lies at bin k = 128. For an unwindowed record, the positive-frequency DFT magnitude is approximately

The one-sided peak-amplitude estimate is therefore

The Nyquist frequency is 4 kHz, so bin 128 corresponds to 1 kHz and is safely within the one-sided range. If the tone were 1003 Hz instead, it would not be bin-centered, and leakage would distribute its energy among neighboring bins. Zero-padding could make the spectral peak appear smoother but would not eliminate leakage.

10. Case Application: Motor-Fault Spectral Monitoring

Background

Consider an illustrative industrial condition-monitoring system for a three-phase induction motor. An accelerometer is mounted near a bearing housing. The objective is to detect a developing bearing defect before vibration becomes severe enough to cause an unplanned shutdown.

Problem

The motor rotates at 1,800 revolutions per minute, corresponding to a shaft frequency of

A bearing-related modulation component is expected near 210 Hz, but the measurement also contains strong components at the shaft frequency, electrical frequencies, and broadband noise. The engineer must distinguish a weak defect-related line from leakage caused by finite records.

Analysis and approach

The accelerometer signal is sampled at 12.8 kHz. A record of N = 4096 samples gives

and

This spacing is sufficient to distinguish a component near 210 Hz from nearby components separated by several hertz, although actual resolution also depends on window main-lobe width and signal-to-noise ratio.

The processing chain is:

• apply an anti-aliasing filter before sampling;
• remove the sample mean and, if justified, a slow trend;
• use a Hann window to reduce sidelobe contamination from strong shaft-related components;
• compute a 4096-point FFT;
• convert the result to a calibrated acceleration spectrum;
• average spectra from successive overlapping records to reduce random variance;
• compare the 210 Hz region with a baseline recorded under comparable load and speed.

If rotational speed changes, the defect frequency may move. In that situation, fixed-frequency averaging can smear the component. Order tracking, which expresses frequency relative to shaft speed, may be more appropriate. For a stationary signal, Welch’s method can provide a lower-variance power spectral density estimate by averaging modified periodograms from overlapping, windowed segments.

Outcome

Suppose the baseline spectrum shows no persistent feature near 210 Hz, whereas later records show a narrow component that increases consistently with operating time. The result does not by itself prove a bearing defect: structural resonances, load variation, sensor mounting, and electrical interference must also be considered. Confirmation may require time-domain impulsiveness measures, envelope analysis, inspection, or a second sensor.

Learning points

The case demonstrates that FFT computation is only one stage of spectral analysis. Reliable diagnosis depends on sampling integrity, frequency resolution, window selection, calibration, statistical averaging, operating-condition control, and engineering interpretation.

11. Spectral Estimation Beyond the Basic DFT

A single periodogram is easy to compute but has high variance. For a stationary random process, the periodogram at frequency f_k is often written in a form proportional to

The exact proportionality depends on sampling interval, record length, window power, and whether a one-sided or two-sided density is required. Because individual periodogram values fluctuate substantially, Welch’s method divides the record into overlapping segments, windows each segment, computes periodograms, and averages them.

Welch averaging reduces variance but generally reduces effective frequency resolution because each segment is shorter than the full record. Multitaper methods use several orthogonal tapers to control leakage and variance more effectively in some applications. Parametric methods, including autoregressive spectral estimation, model the signal statistically and can provide high apparent resolution, but their results depend strongly on model assumptions and order selection.

The DFT is also fundamental to cross-spectral analysis. Given two signals x[n] and y[n], estimates of auto-spectra and cross-spectrum can be constructed from products such as

A commonly used magnitude-squared coherence is

provided the spectral estimates are defined consistently and S_xx(f), S_yy(f) are nonzero. Coherence lies between 0 and 1 for standard averaged spectral estimates and indicates the strength of linear association at each frequency, not causation.

12. Numerical and Implementation Considerations

Finite-precision arithmetic causes round-off error. Floating-point FFTs normally provide adequate accuracy, but long transforms, large dynamic ranges, repeated processing, and fixed-point embedded implementations require scaling and overflow analysis.

Important implementation practices include:

• use a transform length supported efficiently by the selected library;
• document the normalization convention;
• verify inverse-transform reconstruction;
• test Parseval’s relation numerically;
• use known bin-centered tones for amplitude calibration;
• distinguish complex magnitude from magnitude squared;
• avoid applying a one-sided factor of two to DC or Nyquist bins;
• account for window coherent gain and noise bandwidth;
• preserve units through sensor calibration and sampling-rate factors;
• use overlap and averaging consistently in PSD estimation.

For real-valued data, real-to-complex FFT routines reduce storage and computation. In embedded systems, fixed-point FFTs may use stage-by-stage scaling to prevent overflow. The resulting scale factor must be tracked explicitly.

13. Limitations, Governance, and Responsible Interpretation

A spectrum is a representation, not an explanation. Peaks may result from sensor resonance, electromagnetic pickup, aliasing, window artifacts, quantization, or mechanical vibration. Automated spectral classification systems can therefore produce false alarms or miss faults when operating conditions differ from the training data.

Engineering governance should require traceable metadata: sensor identity, calibration date, sampling frequency, anti-aliasing settings, window, transform length, overlap, averaging, units, and operating condition. In safety-critical systems, spectral decisions should be validated against independent measurements and clearly defined alarm thresholds.

When machine learning or AI is used to classify spectra, the model should be evaluated across speed, load, temperature, sensor placement, and equipment variability. Data leakage between training and test records can produce misleadingly high performance. Explainability is particularly important when a spectral alarm triggers maintenance or shutdown decisions.

14. Contemporary Developments

Current computational spectral analysis increasingly combines FFT processing with distributed sensing, edge computing, GPU acceleration, compressed data transport, and adaptive algorithms. Streaming FFT architectures support real-time radio, radar, audio, biomedical monitoring, and industrial diagnostics. Time-frequency reassignment, synchrosqueezing, wavelet methods, and cyclostationary analysis extend conventional Fourier methods for transients, modulation, and rotating machinery.

These methods do not eliminate the fundamental time-frequency trade-off. They provide alternative representations whose assumptions must be matched to the signal and application. The best method is determined by the measurement question, signal stationarity, noise characteristics, computational constraints, and required uncertainty.

Best-Practice Checklist

Before accepting a computational spectrum, verify:

1. The sampling rate is adequate and aliasing is controlled.
2. The record duration meets the required frequency spacing.
3. The DFT length and any zero-padding are clearly distinguished.
4. The window is appropriate for the measurement objective.
5. Amplitude, power, and PSD scaling are not confused.
6. DC, Nyquist, and one-sided corrections are handled correctly.
7. Calibration and physical units are preserved.
8. Leakage, noise variance, and operating-condition changes are considered.
9. Results are validated using known signals or independent evidence.
10. Processing metadata and uncertainty are reported.

Chapter Summary

The DFT converts a finite discrete-time record into a finite set of complex frequency coefficients. Its frequency spacing is Δf = f_s/N = 1/T, while the Nyquist frequency is f_s/2. The DFT represents both samples of a finite-record DTFT and Fourier-series coefficients of a periodically extended sequence.

Finite records produce spectral leakage unless the observed waveform is compatible with periodic extension. Window functions control the trade-off between main-lobe width, sidelobe suppression, and amplitude accuracy. Correct spectrum scaling requires attention to one-sided versus two-sided representations, coherent gain, window power, and equivalent noise bandwidth.

The FFT computes the DFT efficiently by exploiting symmetry and recursively decomposing the transform. Radix-2 decimation-in-time expresses an N-point transform in terms of two N/2-point transforms and butterfly operations, reducing complexity from order N² to order N log₂N.

DFT properties support circular convolution, efficient filtering, correlation, system analysis, and spectral estimation. Zero-padding interpolates the frequency grid but does not increase information or fundamental resolution. For random or nonstationary signals, Welch averaging, multitaper methods, STFTs, and related approaches may be more informative than a single periodogram.

Reliable computational spectral analysis requires a complete measurement chain: anti-aliasing, sampling design, preprocessing, window selection, FFT computation, scaling, validation, and engineering interpretation. The FFT is therefore a powerful computational engine, but sound spectral conclusions depend on theory, calibration, metadata, and critical judgment.

Key Terms

DFT: Finite transform that maps N time-domain samples to N frequency-domain coefficients.

IDFT: Inverse operation that reconstructs the finite sequence from its DFT coefficients.

FFT: Efficient algorithm for computing a DFT.

DTFT: Fourier transform of a discrete-time sequence with generally continuous frequency variable.

Frequency bin: One indexed frequency sample of a DFT.

Nyquist frequency: Half the sampling frequency for a uniformly sampled signal.

Spectral leakage: Spreading of signal energy across frequency bins caused by finite observation and periodic-extension discontinuities.

Window: Finite weighting sequence applied to a data record before transformation.

Main lobe: Central concentration of a window spectrum around a tone.

Sidelobe: Secondary spectral structure away from the main lobe.

Coherent gain: Mean value of a window used to correct bin-centered sinusoidal amplitude.

Equivalent noise bandwidth: Effective bandwidth through which a window passes white-noise power.

Zero-padding: Appending zeros to a record before transformation to obtain a denser frequency grid.

Circular convolution: Convolution under modular indexing, corresponding directly to multiplication of DFTs.

Periodogram: Squared-magnitude spectral estimate based on a finite data record.

Welch’s method: Averaged periodogram method using windowed, usually overlapping segments.

STFT: Sequence of local Fourier transforms used to describe time-varying spectra.

Coherence: Frequency-dependent measure of linear association between two signals.

Aliasing: Irreversible overlap of spectral components caused by insufficient sampling.

Parseval’s relation: Equality relating time-domain energy to appropriately scaled DFT-domain energy.

Review Questions

Short Answer Questions

1. Define the N-point DFT and IDFT, including their normalization convention.
2. What determines DFT frequency spacing, and how is it related to observation duration?
3. Explain why a finite record can produce spectral leakage.
4. Distinguish zero-padding from increasing the actual frequency resolution.
5. Why must coherent gain be considered when measuring the amplitude of a windowed sinusoid?

Long Answer Questions

1. Derive the radix-2 decimation-in-time FFT from the N-point DFT equation.
2. Explain the relationship among the DTFT, DFT, periodic extension, and frequency sampling.
3. Compare rectangular, Hann, Hamming, Blackman, and flat-top windows in terms of main-lobe width, sidelobes, and measurement objectives.
4. Derive the circular-convolution property of the DFT and explain zero-padding requirements for linear convolution.
5. Discuss the design of a reliable computational spectral-analysis workflow for an industrial monitoring system.

Analytical / Application Questions

1. A system samples at 16 kHz and collects 2048 samples. Calculate the observation duration, frequency spacing, and Nyquist frequency. Explain whether zero-padding to 8192 points changes the fundamental resolution.
2. Design a spectral-analysis procedure for detecting a weak vibration component near a strong rotational harmonic. Justify the sampling rate, record length, window, averaging method, and calibration steps.
3. A one-sided spectrum shows a large peak at DC and several low-frequency components. Analyze possible causes and propose preprocessing and validation steps before interpreting the components as physical phenomena.

Multiple Choice Questions with Answers

Question 1

For an N-point DFT sampled at frequency f_s, what is the frequency spacing?

A. f_sN

B. f_s/N

C. N/f_s

D. f_s/2N

Correct Answer: B. f_s/N

Question 2

What is the primary purpose of the FFT?

A. To increase the sampling frequency

B. To remove aliasing after sampling

C. To compute the DFT with lower computational complexity

D. To convert an analog signal directly into a digital signal

Correct Answer: C. To compute the DFT with lower computational complexity

Question 3

A signal is sampled at 8 kHz using N = 1024 samples. What is the DFT frequency spacing?

A. 3.90625 Hz

B. 7.8125 Hz

C. 8 Hz

D. 125 Hz

Correct Answer: B. 7.8125 Hz

Question 4

What does zero-padding before an FFT primarily do?

A. Creates new signal information

B. Raises the Nyquist frequency

C. Produces a denser frequency grid

D. Eliminates spectral leakage

Correct Answer: C. Produces a denser frequency grid

Question 5

Which window generally provides stronger sidelobe suppression than a rectangular window, at the cost of a wider main lobe?

A. Blackman window

B. Impulse window

C. Unit-sample window

D. No window

Correct Answer: A. Blackman window

Question 6

For a real-valued sequence, which DFT relationship expresses conjugate symmetry?

A. X[N−k] = X[k]

B. X[N−k] = −X[k]

C. X[N−k] = X[k]^*

D. X[N−k] = 1/X[k]

Correct Answer: C. X[N−k] = X[k]^*

Question 7

If two finite sequences have lengths L_x and L_h, what minimum zero-padded DFT length guarantees linear convolution through DFT multiplication?

A. max(L_x,L_h)

B. L_x + L_h − 1

C. L_xL_h

D. L_x − L_h

Correct Answer: B. L_x + L_h − 1

Question 8

Which statement about spectral leakage is correct?

A. It occurs only when the sampling frequency is below the Nyquist rate

B. It results partly from discontinuities in the periodic extension of a finite record

C. It is completely removed by zero-padding

D. It affects only the DC bin

Correct Answer: B. It results partly from discontinuities in the periodic extension of a finite record

Question 9

For the normalization used in this chapter, Parseval’s relation includes which factor on the frequency-domain sum?

A. N

B. 1/N

C. √N

D. 2N

Correct Answer: B. 1/N

Question 10

In an STFT, using a shorter analysis window generally gives what trade-off?

A. Better frequency resolution and worse time resolution

B. Better time localization and worse frequency resolution

C. Higher Nyquist frequency and lower sampling rate

D. No change in either time or frequency resolution

Correct Answer: B. Better time localization and worse frequency resolution