Unit 4: The Continuous Time Fourier Transform and Sampling - Subjective Questions

The Continuous Time Fourier Transform (CTFT) is a mathematical tool used to represent an aperiodic continuous-time signal as a continuous sum (integral) of complex exponentials.

Analysis Equation (Forward Transform):

This equation decomposes the signal into its frequency components, giving the spectrum .

Synthesis Equation (Inverse Transform):

This reconstructs the time-domain signal from its frequency components.

Physical Significance:

• is called the spectrum of and is generally a complex function.
• represents the magnitude spectrum (amount of each frequency present).
• represents the phase spectrum (phase of each frequency component).
• The transform bridges the time domain and frequency domain representations of a signal.

Given: ,

Derivation:

Using the analysis equation:

Since for :

Evaluating the integral:

Since , the term vanishes at :

Magnitude Spectrum:

Phase Spectrum:

Sketch Description:

• The magnitude is maximum () at and decreases monotonically as (low-pass nature).
• The phase starts at at , approaches as , and as .

Introduction:
For periodic signals, the ordinary Fourier integral does not converge in the usual sense. However, by using impulse functions, we can express the Fourier Transform of a periodic signal.

Fourier Series Representation:
A periodic signal with fundamental frequency can be written as:

Key Transform Pair:
The Fourier Transform of a complex exponential is:

Derivation:
Taking the Fourier Transform of both sides of the Fourier series:

Interchanging summation and integration:

Result:

Conclusion:

• The Fourier Transform of a periodic signal consists of a train of impulses located at integer multiples of .
• The strength (area) of each impulse is , proportional to the corresponding Fourier series coefficient.

1. Linearity Property:

If and , then:

Proof:

2. Time Shifting Property:

If , then:

Proof:

Let , :

Significance: A time shift does not affect the magnitude spectrum but introduces a linear phase shift .

Convolution Property:

If and , then:

Proof:
The convolution is defined as:

Taking the Fourier Transform:

Interchanging the order of integration:

Using the time-shifting property, the inner integral equals :

Importance in LTI Systems:

• Convolution in the time domain becomes simple multiplication in the frequency domain.
• is called the frequency response of the system.
• It greatly simplifies analysis: the output spectrum is obtained by multiplying the input spectrum with the system frequency response.
• It forms the basis of filter design and system characterization.

Parseval's Relation:

For an energy signal with Fourier Transform :

Proof:
The total energy is:

Expressing via the inverse transform:

Substituting:

Interchanging the order of integration:

Physical Interpretation:

• The total energy of a signal can be computed in either the time or frequency domain.
• is called the energy spectral density, representing energy distribution across frequencies.
• Energy is conserved between the two domains.

1. Frequency Shifting (Modulation) Property:

If , then:

Proof:

Significance: Multiplying by a complex exponential shifts the spectrum by . This is the basis of amplitude modulation in communication systems.

2. Time Scaling Property:

If , then:

Proof (for ):

Let , :

For general , the factor is .

Significance: Compression in time () causes expansion in frequency and vice versa — the time-bandwidth inverse relationship.

Sampling Theorem (Nyquist-Shannon):

A band-limited continuous-time signal having no frequency components higher than Hz (i.e., for ) can be completely represented and recovered from its samples if the sampling frequency satisfies:

Nyquist Rate:

• The minimum sampling rate required to avoid loss of information (aliasing).
  

Nyquist Interval:

• The maximum time interval between successive samples.
  

Explanation:

• If sampling is done at exactly or above the Nyquist rate, the spectral replicas do not overlap, allowing perfect reconstruction using an ideal low-pass filter.
• If , aliasing occurs and the original signal cannot be recovered.

Example: For a speech signal band-limited to kHz, the Nyquist rate is kHz and the Nyquist interval is .

Impulse Train Sampling:

The sampling is modeled as multiplication of with a periodic impulse train:

where is the sampling period.

Sampled Signal:

Spectrum Derivation:

The Fourier Transform of the impulse train is another impulse train:

Since multiplication in time = convolution in frequency (scaled by ):

Formation of Spectral Replicas:

• The spectrum of the sampled signal consists of shifted replicas of the original spectrum , centered at integer multiples of .
• Each replica is scaled by .
• If , the replicas do not overlap, and the original signal can be recovered.
• If , replicas overlap causing aliasing.

Reconstruction Concept:

To recover from its samples , the sampled signal is passed through an ideal low-pass filter (LPF) with cutoff frequency (where ).

Ideal LPF Characteristics:

The impulse response of an ideal LPF (with ) is a sinc function:

Interpolation Formula Derivation:

The reconstructed signal is the convolution of with :

Interpretation:

• Each sample is replaced by a sinc pulse centered at and weighted by the sample value.
• The sum of all these sinc functions reconstructs the original continuous signal.
• This is called band-limited (ideal) interpolation.

Aliasing:

Aliasing is the phenomenon in which high-frequency components of a signal take on the identity of lower-frequency components after sampling. It occurs when a signal is sampled at a rate below the Nyquist rate ().

Cause – Undersampling:

When sampling, the spectrum of the sampled signal is:

• If : replicas are well separated — no aliasing.
• If : adjacent replicas overlap in the region near .

Spectral Diagram Description:

• With proper sampling, distinct triangular spectra sit at with gaps between them.
• With undersampling, the tails of neighbouring triangles overlap and add up, distorting the spectrum in the overlap region.

Effects of Aliasing:

• The original spectrum is corrupted and cannot be perfectly recovered.
• High frequencies appear as spurious low frequencies (frequency folding).
• Reconstructed signal is distorted.

Prevention:

• Sample at .
• Use an anti-aliasing (pre-sampling) low-pass filter to band-limit the signal before sampling.

Given:

Derivation:

Using :

Comments on Spectrum:

• The spectrum is a sinc function, which is real and even.
• The main lobe width is inversely proportional to (wider pulse → narrower spectrum).
• Zero crossings occur at for non-zero integer .
• This illustrates the time-frequency duality: a rectangular pulse in time gives a sinc in frequency.

1. Differentiation in Time Property:

If , then:

Proof:
From the inverse transform:

Differentiating both sides with respect to :

This shows that the transform of is .

2. Integration Property:

If , then:

Explanation:

• Integration corresponds to division by in frequency.
• The additional term accounts for the possible DC (average) value of the integrated signal.

Significance:

• Differentiation amplifies high frequencies (multiplication by ).
• Integration attenuates high frequencies and emphasizes low frequencies (division by ).

Comparison of Fourier Series (FS) and Fourier Transform (FT):

When to use:

• Fourier Series: Used to analyze the frequency content of periodic signals, giving a set of discrete harmonic coefficients.
• Fourier Transform: Used for aperiodic (finite-energy) signals to obtain a continuous frequency spectrum.

Relationship:

• As the period , a periodic signal becomes aperiodic and the discrete FS spectrum merges into the continuous FT spectrum.

Duality Property:

If , then:

This property exploits the symmetry between the analysis and synthesis equations of the Fourier Transform.

Proof:
From the inverse transform:

Replace with :

Now interchange the roles of and (rename variables):

Hence .

Example:

• We know a rectangular pulse in time gives a sinc in frequency.
• By duality, a sinc function in time gives a rectangular (ideal low-pass) spectrum in frequency.
• This is why the impulse response of an ideal LPF is a sinc function.

Significance: Duality allows us to derive new transform pairs from known ones without recomputing integrals.

Software Simulation of Frequency Spectrum:

Real-world signals (speech, audio, ECG, vibration, etc.) are continuous. To analyze their spectrum on a computer, the following steps are followed:

Steps Involved:

1. Acquisition & Sampling: The continuous signal is sampled at rate (Nyquist criterion) to obtain a discrete sequence .
2. Anti-Aliasing Filtering: A low-pass filter is applied before sampling to remove frequencies above and prevent aliasing.
3. Windowing: A finite segment of samples is selected using a window (rectangular, Hamming, Hanning) to reduce spectral leakage.
4. Apply DFT/FFT: The Discrete Fourier Transform computes the spectrum:
   
   The Fast Fourier Transform (FFT) is an efficient algorithm to compute the DFT in operations.
5. Plot Magnitude & Phase: vs frequency gives the magnitude spectrum.

Tools: MATLAB (fft), Python (NumPy/SciPy numpy.fft.fft), Octave, etc.

Role of DFT/FFT:

• Converts a finite discrete-time sequence into a discrete frequency representation.
• FFT drastically reduces computation time, enabling real-time analysis.

Effect of Sampling:

• Frequency resolution is .
• Insufficient sampling causes aliasing; too few samples reduce resolution.
• Larger → finer frequency resolution.

Given:

• Signal frequency: Hz
• Sampling frequency: Hz

Step 1 – Check Nyquist Criterion:

Nyquist rate Hz.

Since , the signal is undersampled ⟹ aliasing occurs.

Step 2 – Determine Aliased Frequency:

The folding frequency is Hz. When , the apparent (aliased) frequency is:

Step 3 – Verification:

Since Hz Hz, this is a valid aliased frequency lying within the baseband.

Result:

• Aliasing occurs.
• The Hz component appears as a Hz component after sampling.

Conclusion: To avoid this, the sampling frequency must be at least Hz, or an anti-aliasing filter should band-limit the input below Hz.

Software Simulation of Undersampling:

Undersampling and its effect (aliasing) can be clearly demonstrated using MATLAB or Python.

Simulation Procedure:

1. Generate a high-resolution reference signal, e.g. with Hz, using a very high sampling rate (to approximate the continuous signal).
2. Sample the signal at two rates:
   • Adequate: Hz (satisfies Nyquist).
   • Insufficient: Hz (below Hz).
3. Reconstruct/interpolate and plot both cases.
4. Compute the FFT of both sampled signals and observe the spectrum.

Sample Python Pseudocode:
python
import numpy as np
import matplotlib.pyplot as plt

f0 = 100 # signal frequency
t = np.arange(0, 0.05, 1/10000) # 'continuous' time
x = np.sin(2np.pif0*t)

fs = 120 # undersampled rate
ts = np.arange(0, 0.05, 1/fs)
xs = np.sin(2np.pif0*ts)

plt.plot(t, x, label='Original')
plt.stem(ts, xs, 'r', label='Undersampled')
plt.legend(); plt.show()

Observations:

• With adequate sampling, the samples clearly trace the original waveform; FFT shows a peak at Hz.
• With undersampling, the samples appear to follow a lower-frequency waveform; FFT shows a peak at the aliased frequency Hz.

Conclusion: The simulation visually confirms that undersampling creates a false (alias) low-frequency signal, demonstrating why the Nyquist criterion must be met.

Given:

Applying the Inverse Fourier Transform:

Term 1:

Term 2:

Term 3:

Combining:

Using Euler's identity :

Identification:

• The signal is a DC component of value 1 plus a cosine of frequency .
• The impulse at corresponds to the DC term; the pair of impulses at correspond to the cosine.

Types of Sampling:

1. Ideal (Impulse) Sampling:

• Signal is multiplied by an ideal impulse train .
• Samples exist only at instants with zero width.
  
• Advantage: Simplest for mathematical analysis; spectrum is perfectly periodic replicas of .
• Limitation: Physically not realizable (ideal impulses cannot be generated).

2. Natural Sampling:

• Signal is multiplied by a pulse train of finite width .
• The top of each sample follows the signal shape during pulse duration.
  
• Advantage: Practically realizable using analog switches.
• Spectrum: Replicas are weighted by a sinc envelope but original shape within each pulse is preserved.

3. Flat-Top Sampling (Sample-and-Hold):

• Each sample value is held constant for the pulse duration (rectangular flat top).
• Advantage: Easy to implement with sample-and-hold circuits; used in practical ADCs.
• Limitation: Introduces aperture effect — high-frequency attenuation described by a sinc distortion, requiring equalization during reconstruction.

Comparison Table:

Conclusion: Ideal sampling is theoretical; flat-top sampling is the most common in real ADC systems, though it needs correction for the aperture effect.