4. Continuous-Time and Discrete-Time Fourier Transforms

Introduction and Context

Fourier analysis provides one of the most powerful ways of studying signals and linear time-invariant (LTI) systems. A signal described in the time domain specifies how a quantity changes with time; its Fourier representation describes the same signal in terms of complex sinusoids of different frequencies. This change of representation is not merely mathematical. It enables engineers to identify bandwidth, design filters, analyze modulation, evaluate distortion, interpret measurements, and determine how systems alter signal spectra.

Two transforms are central to this chapter. The continuous-time Fourier transform (CTFT) is used for continuous-time signals, whereas the discrete-time Fourier transform (DTFT) is used for discrete-time sequences. They are closely related but not identical. In particular, the CTFT is generally continuous in frequency and may be nonperiodic, while the DTFT is periodic in frequency with period 2π radians per sample.

Fourier transforms are especially useful because convolution in time becomes multiplication in frequency. Thus, the output spectrum of an LTI system is obtained by multiplying the input spectrum by the system frequency response. The transform domain also makes modulation, sampling-related spectral repetition, energy calculations, and bandwidth estimation more transparent.

Learning Objectives

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

1. Define the CTFT and DTFT and explain their mathematical and physical interpretations.

2. Distinguish energy spectra, power spectra, magnitude spectra, and phase spectra.

3. Apply linearity, time shifting, frequency shifting, scaling, differentiation, convolution, and multiplication properties.

4. Analyze amplitude and frequency modulation using spectral translation.

5. Determine the frequency response and output spectrum of an LTI system.

6. Use Parseval relations to calculate signal energy or average power.

7. Estimate bandwidth and assess amplitude, phase, and delay distortion.

8. Recognize practical limitations associated with finite records, leakage, aliasing, and numerical computation.

9. Select appropriate Fourier-domain methods in engineering applications.

From Sinusoidal Decomposition to Transform Representations

A complex exponential is written as

x(t) = Ae^{j\omega t},

where A is a complex amplitude and ω is angular frequency in radians per second. Complex exponentials are eigenfunctions of LTI systems: if the input is e^{jωt}, the output is H(jω)e^{jωt}, where H(jω) is the system frequency response. This property explains the central role of Fourier analysis.

A general finite-energy signal can be represented as a continuous superposition of complex exponentials. The CTFT pair is defined by

X(jω) = ∫_{−∞}^{∞} x(t)e^{−jωt}dt,

x(t) = 1/(2π) ∫_{−∞}^{∞} X(jω)e^{jωt}dω.

Here X(jω) is the Fourier transform of x(t), and the second equation is the inverse Fourier transform. The transform variable ω is angular frequency. If ordinary frequency f is used, then ω = 2πf.

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

X(e^{jω}) = Σ_{n=−∞}^{∞} x[n]e^{−jωn},

with inverse relation

x[n] = 1/(2π) ∫_{−π}^{π} X(e^{jω})e^{jωn}dω.

The notation X(e^{jω}) emphasizes that the DTFT is a function of a point on the unit circle in the z-plane. Since e^{j(ω+2π)n} = e^{jωn} for every integer n,

X(e^{j(ω+2π)}) = X(e^{jω}).

Consequently, the DTFT is periodic with period 2π. It is sufficient to specify it over any interval of length 2π, commonly −π ≤ ω < π.

The difference between the two transforms follows from the nature of time. Continuous time permits arbitrary real t, whereas discrete time restricts t to integer indices. The sampling of time produces periodicity in the frequency variable. This duality is fundamental: discretizing one domain creates periodic structure in the other.

Important Spectral Concepts

The transform is generally complex and may be expressed as

X(jω) = |X(jω)|e^{j∠X(jω)}

for CTFTs, or

X(e^{jω}) = |X(e^{jω})|e^{j∠X(e^{jω})}

for DTFTs. The magnitude spectrum describes the amplitude associated with each frequency, while the phase spectrum describes the relative phase of its components. Two signals can have identical magnitude spectra but different waveforms because their phase spectra differ.

The energy spectral density of a finite-energy signal is |X(jω)|² for the CTFT, with the appropriate 1/(2π) factor when integrated to obtain total energy. For discrete-time signals, the corresponding quantity is |X(e^{jω})|². A power spectral density is used for power signals, for which total energy is infinite but average power is finite. Strictly speaking, power spectral density is often defined through limiting procedures or autocorrelation functions rather than by directly squaring an ordinary Fourier transform.

A signal is band-limited if its spectrum is zero outside a finite frequency interval. The smallest such interval, when it exists, defines the essential bandwidth. Practical signals are often not exactly band-limited, so bandwidth may be specified using a threshold, occupied-power percentage, or a first-null criterion.

For real-valued x(t), the CTFT has conjugate symmetry:

X(jω) = X*(−jω).

The same property holds for real sequences in the DTFT. Therefore, the magnitude spectrum is even and the phase spectrum is odd, subject to phase wrapping and phase discontinuities at spectral zeros.

Transform Pairs and Properties

A small number of transform pairs, combined with the transform properties, solve a large class of engineering problems.

For the CTFT, common pairs include:

• δ(t) ↔ 1.
• 1 ↔ 2πδ(ω).
• e^{jω₀t} ↔ 2πδ(ω−ω₀).
• cos(ω₀t) ↔ π[δ(ω−ω₀)+δ(ω+ω₀)].
• sin(ω₀t) ↔ (π/j)[δ(ω−ω₀)−δ(ω+ω₀)].
• e^{−at}u(t), a > 0 ↔ 1/(a+jω).
• rect(t/T) ↔ T sinc(ωT/2), where sinc(z) = sin(z)/z.

For the DTFT:

• δ[n] ↔ 1.
• δ[n−n₀] ↔ e^{−jωn₀}.
• a^n u[n], |a| < 1 ↔ 1/(1−ae^{−jω}).
• e^{jω₀n} ↔ 2πΣ_k δ(ω−ω₀−2πk), interpreted periodically.
• cos(ω₀n) ↔ πΣ_k[δ(ω−ω₀−2πk)+δ(ω+ω₀−2πk)].

Linearity

If x₁(t) ↔ X₁(jω) and x₂(t) ↔ X₂(jω), then

ax₁(t)+bx₂(t) ↔ aX₁(jω)+bX₂(jω).

The same relation applies to DTFTs. Linearity allows complicated signals to be decomposed into simpler components.

Time Shifting

For the CTFT,

x(t−t₀) ↔ e^{−jωt₀}X(jω).

For the DTFT,

x[n−n₀] ↔ e^{−jωn₀}X(e^{jω}).

A time shift changes phase but not magnitude. This is important in communication and measurement systems: delay can alter waveform alignment without changing spectral magnitude.

Frequency Shifting and Modulation

Multiplication by a complex exponential shifts a spectrum:

x(t)e^{jω₀t} ↔ X(j(ω−ω₀)),

x[n]e^{jω₀n} ↔ X(e^{j(ω−ω₀)}).

For multiplication by a cosine,

x(t)cos(ω₀t) ↔ 1/2[X(j(ω−ω₀))+X(j(ω+ω₀))].

Thus, cosine modulation produces two translated copies of the original spectrum. This result forms the spectral basis of amplitude modulation, mixers, and heterodyne receivers.

Time Scaling

For a ≠ 0,

x(at) ↔ 1/|a| X(jω/a).

Compression in time expands the spectrum, while expansion in time compresses it. The factor 1/|a| preserves the relationship between the time and frequency representations.

Differentiation and Integration

Differentiation in time corresponds to multiplication by jω:

(d/dt)x(t) ↔ jωX(jω).

For sequences, a useful difference relation is

x[n]−x[n−1] ↔ (1−e^{−jω})X(e^{jω}).

These properties explain why differentiators emphasize high frequencies and why discrete differences act as high-pass operations.

Convolution and Multiplication

If y(t) = x(t)*h(t), then

Y(jω) = X(jω)H(jω).

Similarly, for discrete time,

y[n] = x[n]*h[n] ↔ Y(e^{jω}) = X(e^{jω})H(e^{jω}).

Convolution in time is multiplication in frequency. Conversely, multiplication in time corresponds to convolution in frequency, with scaling determined by the transform convention. This duality is central to filtering and windowing.

LTI Systems, Frequency Response, and Distortion

For a continuous-time LTI system with impulse response h(t),

y(t)=x(t)*h(t),

and therefore

Y(jω)=H(jω)X(jω), H(jω)=\mathcal{F}\{h(t)\}.

The discrete-time relation is

Y(e^{jω})=H(e^{jω})X(e^{jω}).

The magnitude |H| determines amplitude modification, and the phase ∠H determines phase modification. A system introduces no amplitude distortion over a signal band if |H(jω)| is constant over that band. It introduces no waveform distortion, apart from an overall gain and delay, if

H(jω)=Ke^{−jωt₀}

within the occupied band. The corresponding discrete-time condition is H(e^{jω}) = Ke^{−jωn₀}.

The group delay is

τ_g(ω)=−d∠H(jω)/dω.

A constant group delay means that all frequency components experience the same delay. Variation in group delay produces phase or delay distortion even when amplitude response is relatively flat. In data communication, group-delay variation can cause intersymbol interference and degrade error performance.

For a first-order low-pass system,

H(jω)=1/(1+jωτ).

Its magnitude decreases as frequency increases, and its phase approaches −π/2 at high frequencies. A sinusoidal input at frequency ω₀ emerges at the same frequency, with amplitude scaled by |H(jω₀)| and phase shifted by ∠H(jω₀). No new frequencies are created by an LTI system; it only weights existing spectral components.

Parseval Relations and Signal Measurements

Parseval’s relation establishes equality between time-domain energy and frequency-domain energy. For a continuous-time finite-energy signal,

∫_{−∞}^{∞}|x(t)|²dt = 1/(2π)∫_{−∞}^{∞}|X(jω)|²dω.

For a discrete-time finite-energy sequence,

Σ_{n=−∞}^{∞}|x[n]|² = 1/(2π)∫_{−π}^{π}|X(e^{jω})|²dω.

These formulas are useful when the spectral representation is easier to integrate or measure than the waveform. For example, the energy contained in a frequency band B is obtained by integrating the energy spectral density only over B.

For a periodic continuous-time signal with Fourier-series coefficients C_k, average power is Σ_k|C_k|². A related discrete-time result applies to periodic sequences. Fourier transforms of periodic signals contain impulses rather than ordinary continuous functions, so Fourier-series and distribution concepts must be used carefully.

Modulation and Spectral Translation

Consider a baseband signal m(t) with spectrum M(jω). If it is multiplied by cos(ω_ct), the modulated signal is

s(t)=m(t)cos(ω_ct),

and

S(jω)=1/2[M(j(ω−ω_c))+M(j(ω+ω_c))].

The baseband spectrum is translated to neighborhoods around ±ω_c. If m(t) is band-limited to B radians per second and ω_c>B, the positive- and negative-frequency bands do not overlap. This is the spectral basis of double-sideband amplitude modulation.

In coherent demodulation, multiplication by another cosine translates the modulated spectrum back toward baseband. A low-pass filter then removes high-frequency terms. The filter cutoff must preserve the desired baseband while rejecting the translated unwanted components.

Single-sideband transmission suppresses one of the translated sidebands, reducing bandwidth and power requirements. In practical systems, imperfect filters and carrier or phase errors cause residual interference, so spectral analysis must include transition bands and implementation tolerances.

Discrete-Time Spectra and Digital Filtering

The DTFT is particularly useful for analyzing digital filters. If a filter has difference equation

y[n]=Σ_{k=0}^{M}b_kx[n−k]−Σ_{k=1}^{N}a_ky[n−k],

then, under appropriate initial-rest conditions,

H(e^{jω})=Y(e^{jω})/X(e^{jω})

= [Σ_{k=0}^{M}b_ke^{−jωk}]/[1+Σ_{k=1}^{N}a_ke^{−jωk}].

The response is periodic, so frequencies separated by integer multiples of 2π are indistinguishable in discrete time. A digital low-pass filter therefore passes frequencies near ω=0 and rejects frequencies near ±π, while a high-pass filter often does the reverse.

A moving-average filter illustrates the relation between implementation and spectrum:

y[n]=1/L Σ_{k=0}^{L−1}x[n−k].

Its frequency response is

H(e^{jω})=1/L Σ_{k=0}^{L−1}e^{−jωk}.

The response has a main lobe around zero frequency and nulls at frequencies related to 2π/L. Increasing L produces stronger smoothing but also narrows the passband and increases delay.

Fourier Analysis in Practice

In measurement systems, signals are rarely available over infinite time. A finite record x[n], n=0,…,N−1, is often transformed using the discrete Fourier transform (DFT):

X[k]=Σ_{n=0}^{N−1}x[n]e^{−j2πkn/N}.

The DFT samples the DTFT at ω_k=2πk/N, provided the finite record is treated as one period of a periodic sequence. The fast Fourier transform (FFT) is an efficient algorithm for computing the DFT; it is not a different transform.

Finite observation is equivalent to multiplying the underlying signal by a window. Multiplication in time convolves the spectrum with the window spectrum. This creates spectral leakage when the record does not contain an integer number of cycles. Window functions such as Hann, Hamming, and Blackman reduce sidelobes but broaden the main lobe. The choice is therefore a trade-off between frequency resolution and leakage suppression.

Frequency resolution is approximately f_s/N for an N-point record sampled at rate f_s. Zero padding creates more interpolated frequency samples and improves visual smoothness, but it does not increase the fundamental resolving power determined by record duration and window characteristics.

In sampled systems, the sampling frequency must be sufficiently high to prevent spectral overlap caused by aliasing. The CTFT spectrum of a sampled signal consists of shifted replicas spaced by the sampling angular frequency. An anti-aliasing low-pass filter is therefore required before analog-to-digital conversion when the input may contain frequencies above half the sampling rate.

Applied Case Study: Diagnosing Vibration in a Motor-Drive Assembly

Background

Consider an illustrative industrial case involving a three-phase motor coupled to a gearbox and a variable-frequency drive. Maintenance personnel observe increasing vibration and a tonal acoustic component during operation. Time-domain measurements show a complicated oscillatory waveform, but the source is not immediately apparent.

Situation and Problem

An accelerometer samples the vibration at 10 kHz for 2.048 s, producing 20,480 samples. The motor rotates at 1,500 revolutions per minute, corresponding to 25 Hz. Gear-mesh frequencies are expected near several hundred hertz. The engineering question is whether the observed vibration is caused by imbalance, shaft misalignment, electrical excitation, or a developing gear defect.

Analysis

The record is detrended and multiplied by a Hann window before FFT computation. The frequency spacing is approximately 10,000/20,480 = 0.488 Hz. The spectrum shows a strong component near 25 Hz, harmonics near 50 and 75 Hz, and sidebands around a higher-frequency gear-mesh component. The 25 Hz component is consistent with rotational imbalance. Harmonics suggest nonideal mechanical behavior, while symmetrical sidebands around the gear-mesh frequency indicate modulation by shaft rotation.

The sideband interpretation follows the modulation property: a periodic mechanical variation multiplies or modulates the higher-frequency gear vibration, producing translated spectral components. A time-domain inspection alone would not readily identify these relationships.

Approach and Solution

The maintenance team compares spectra from several operating loads and tracks the amplitude of the 25 Hz component and the gear-mesh sidebands. A narrow-band digital filter isolates the gear-mesh region, while a low-frequency filter monitors rotational components. The filter phase response is checked because phase distortion could alter transient waveform interpretation. Measurements are repeated with a longer record when closely spaced sidebands must be resolved.

Outcome

The rotational line remains stable, but the gear-mesh sidebands increase with load. Inspection identifies wear on a gearbox tooth. The motor itself is not replaced; instead, the gearbox is scheduled for controlled maintenance.

Learning Points

The case demonstrates that spectral peaks must be interpreted in relation to mechanical frequencies, bandwidth, windowing, and system response. A peak is not automatically evidence of a fault. Its location, harmonics, sidebands, dependence on operating conditions, and measurement chain must all be considered.

Limitations, Risks, and Common Errors

Fourier analysis is powerful but not automatic. Several errors occur frequently.

First, transform conventions differ. Some texts place 1/(2π) in the forward transform rather than the inverse transform. Calculations are valid only when one convention is used consistently.

Second, angular frequency and ordinary frequency are often confused. The relations ω=2πf and dω=2πdf must be respected when changing variables.

Third, impulse spectra and ordinary functions should not be treated identically. Periodic sinusoids have line spectra represented by impulses, whereas finite-energy aperiodic signals generally have continuous spectra.

Fourth, a magnitude-only spectrum does not uniquely specify a signal. Discarding phase can destroy timing information and conceal waveform distortion.

Fifth, FFT peaks can be misinterpreted because of leakage, finite resolution, window effects, noise, and inadequate calibration. Averaging may reduce random variance, but it cannot correct aliasing or an inappropriate sampling rate.

Sixth, spectral components introduced by nonlinear devices are not explained by LTI filtering alone. Nonlinearity can create harmonics and intermodulation products. An engineer must first establish whether the assumed linear model is appropriate.

Professional and Ethical Considerations

Spectral measurements can influence safety decisions, maintenance schedules, and product acceptance. Engineers should document sampling rate, sensor characteristics, calibration, window, record length, filtering, and uncertainty. A visually attractive spectrum should not be presented as conclusive evidence when measurement limitations are unknown.

In safety-critical systems, alarm thresholds should be validated against operating conditions and false-alarm consequences. Data integrity is also important: preprocessing choices must be reproducible, and anomalous records should not be selectively omitted without justification. When automated or AI-assisted condition-monitoring tools classify spectral patterns, their training range, confidence, failure modes, and human oversight requirements must be understood.

Current Developments and Best Practices

Modern spectral analysis increasingly combines Fourier methods with time-frequency techniques. Short-time Fourier transforms track changing spectra by applying a sequence of local windows. Wavelet and related methods provide improved localization for transients and multiscale phenomena, although they do not replace the CTFT or DTFT for stationary analysis.

Best practice includes the following:

• Define the engineering question before selecting a transform or window.
• Use units consistently and state whether frequency is in hertz or radians per second.
• Inspect both magnitude and phase when delay or waveform fidelity matters.
• Verify sampling adequacy and apply anti-alias filtering.
• Report record length, frequency resolution, window type, and normalization.
• Compare spectral results with a physical model and independent measurements.
• Distinguish deterministic lines, broadband noise, leakage, and artifacts.
• Use the DFT or FFT as a computational approximation to the DTFT, not as a reason to ignore its assumptions.

Chapter Summary

The CTFT represents continuous-time signals as continuous distributions of complex exponentials, while the DTFT represents discrete-time sequences and is periodic in frequency. Magnitude and phase together describe the spectral content of a signal. Linearity, shifting, scaling, modulation, differentiation, convolution, and multiplication properties provide practical methods for deriving transforms and predicting system behavior.

For LTI systems, the output spectrum is the product of the input spectrum and frequency response. Amplitude distortion is associated with variation in magnitude response, while phase and group-delay variation cause waveform distortion. Parseval relations connect time-domain energy with frequency-domain energy. Fourier analysis also explains modulation, digital filtering, bandwidth, vibration diagnosis, and spectral measurement.

In practical FFT-based analysis, finite records, windowing, leakage, frequency resolution, normalization, and aliasing must be considered. Sound engineering judgment requires interpreting spectral evidence in the context of physical models, instrumentation, uncertainty, and professional responsibility.

Key Terms

CTFT: Fourier transform relating a continuous-time signal to its frequency representation.

DTFT: Fourier transform of a discrete-time sequence, periodic in frequency with period 2π.

Frequency response: Ratio of output spectrum to input spectrum for an LTI system.

Magnitude spectrum: Absolute value of a complex Fourier transform.

Phase spectrum: Argument or phase angle of a Fourier transform.

Bandwidth: Frequency interval containing the significant spectral content of a signal.

Convolution: Operation that combines two signals and becomes multiplication under Fourier transformation.

Modulation: Multiplication or other operation that translates spectral content to different frequencies.

Parseval relation: Equality between signal energy or power calculated in time and frequency domains.

Energy spectral density: Distribution of signal energy over frequency.

Power spectral density: Distribution of average signal power over frequency.

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

Amplitude distortion: Unwanted variation of system gain across the signal band.

Phase distortion: Unwanted non-linear phase or group-delay variation across the signal band.

DFT: Finite-length, discrete-frequency transform used for numerical spectral computation.

FFT: Efficient algorithm for calculating the DFT.

Spectral leakage: Spreading of spectral energy caused mainly by finite observation and windowing.

Aliasing: Irreversible spectral overlap caused by insufficient sampling rate.

Window function: Finite-time weighting function used before spectral estimation.

LTI system: Linear time-invariant system characterized by convolution and frequency response.

Review Questions

Short Answer Questions

1. Why is the DTFT periodic in frequency while the CTFT is not generally periodic?

2. State the CTFT time-shifting property and explain its effect on magnitude and phase.

3. What is the physical meaning of a system frequency response?

4. State the discrete-time Parseval relation.

5. Distinguish spectral leakage from aliasing.

Long Answer Questions

1. Derive the Fourier transform relationship for multiplication of a signal by a cosine and explain its role in amplitude modulation.

2. Explain how convolution, frequency response, and filtering are related for continuous-time and discrete-time LTI systems.

3. Discuss amplitude distortion, phase distortion, and group delay, including the condition for distortionless transmission.

4. Explain the practical computation of a spectrum using the DFT and FFT, including record length, windowing, and frequency resolution.

5. Compare CTFT and DTFT representations with respect to signal domain, frequency variable, periodicity, and engineering applications.

Analytical / Application Questions

1. A signal has spectrum confined to |ω|≤B and is multiplied by cos(ω_ct), where ω_c>B. Determine the locations and bandwidths of the translated spectral copies and state the condition under which they do not overlap.

2. A digital moving-average filter has length L. Explain qualitatively how increasing L changes smoothing, bandwidth, null locations, and delay.

3. An FFT spectrum contains a broad cluster of peaks around a known gear-mesh frequency, with sidebands spaced by the shaft rotation frequency. Use Fourier properties to propose a physical interpretation and describe additional measurements needed to validate it.

22. Multiple Choice Questions with Answers

Question 1

Which statement correctly describes the DTFT?

A. It is nonperiodic for every discrete-time sequence

B. It is periodic in angular frequency with period 2π

C. It exists only for periodic sequences

D. It has frequency measured only in hertz

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

Question 2

A delay of t₀ seconds applied to a continuous-time signal changes its Fourier transform by which factor?

A. e^{jωt₀}

B. e^{−jωt₀}

C. e^{jω/t₀}

D. 1/t₀

Correct Answer: B. e^{−jωt₀}

Question 3

If y(t)=x(t)*h(t), which relation is correct?

A. Y(jω)=X(jω)+H(jω)

B. Y(jω)=X(jω)−H(jω)

C. Y(jω)=X(jω)H(jω)

D. Y(jω)=X(jω)/H(jω) for all systems

Correct Answer: C. Y(jω)=X(jω)H(jω)

Question 4

Multiplication of x(t) by cos(ω_ct) produces

A. A time delay only

B. Two frequency-shifted copies of X(jω), scaled by one-half

C. Differentiation of X(jω)

D. A spectrum compressed by ω_c

Correct Answer: B. Two frequency-shifted copies of X(jω), scaled by one-half

Question 5

For a distortionless LTI system over a signal band, the frequency response should have the form

A. Kω

B. Ke^{−jωt₀}

C. K e^{jω²}

D. 1/(1+jω)

Correct Answer: B. Ke^{−jωt₀}

Question 6

Which relation gives the energy of a finite-energy continuous-time signal in the frequency domain?

A. ∫|x(t)|dt = ∫|X(jω)|dω

B. ∫|x(t)|²dt = (1/2π)∫|X(jω)|²dω

C. ∫x(t)dt = ∫X(jω)dω

D. ∫|x(t)|²dt = 2π∫|X(jω)|²dω

Correct Answer: B. ∫|x(t)|²dt = (1/2π)∫|X(jω)|²dω

Question 7

A finite record does not contain an integer number of cycles of a sinusoid. The most likely spectral consequence is

A. Aliasing is automatically eliminated

B. Spectral leakage

C. The signal becomes noncausal

D. The frequency response becomes linear

Correct Answer: B. Spectral leakage

Question 8

For an N-point record sampled at f_s, the approximate DFT frequency spacing is

A. Nf_s

B. f_s/N

C. N/f_s

D. 2πNf_s

Correct Answer: B. f_s/N

Question 9

What is the principal purpose of an anti-aliasing filter before analog-to-digital conversion?

A. To increase record length

B. To prevent spectral overlap from frequencies above the allowable band

C. To add harmonics to the input

D. To make all phase responses linear

Correct Answer: B. To prevent spectral overlap from frequencies above the allowable band

Question 10

A digital filter has a frequency response whose phase changes nonlinearly across the signal band while its magnitude is nearly constant. The likely impairment is

A. Amplitude clipping only

B. Phase or delay distortion

C. Sampling-rate expansion

D. Perfect distortionless transmission

Correct Answer: B. Phase or delay distortion