Fourier Transform

A non periodic Signal and Periodic signal relationship is based on

A non periodic Signal and Periodic signal relationship is based on

Period

Increasing the period to infinity changes a periodic signal to a non periodic signal

Repeating a non periodic signal with a period changes it to a periodic signal

A Fourier Transforms relationship with Fourier series is

Finding the Fourier series of a periodic signal with its period tending to infinity

What happens to the Fourier Series as the period T increases

• The Spectrum Shape remains the same

• The Spectrum gets denser i.e frequency spacing gets smaller

• The magnitude of the spectral lines decrease

Forward transforms zero crossings of a pulse is dependent on the period T or F

False its dependent on the width of the pulse and multiples f = m/tao

Effect of Pulse width using the Forward FT

The spectrum of it Forward FT shrinks with an increase in the pulse of the signal

For a constant signal For.FT i.e tao of a pulse → 0

A impulse in the FT

Shifting a signal in time changes

Linearly the phase of its spectrum and the magnitude is not affected

real signal spectrum displays Hermitian symmetry t or f

true - symmetric amplitude and odd symmetric phase

Fourier Transform of periodic signals is the

Fourier series

The A/D Converter comprises of

Sampler, Quantizer and Coder

The sample ensures it output signal has discrete values T or F

False, The sampler ensure Discrete time while the quantizer ensures discrete values - Quantized signal.

What does the Coder do

It encodes the quantized signal to digital input, 0101. It can use a variety of codes to represents quantized signal

Why is the choice of the sampling frequency important

To prevent Overlapping .

if w_s >= 2w_n , w_s - w_n >= 2w_n- w_n → w_s - w_n >= w_n . This ensures no overlapping within the spectrum . if w_s <= 2w_n then w_s - w_n <= w_n → Overlapping

A phenomenon where instead of original copy of the input when reconstruction occurs, A different spectrum is received.

Aliasing. Occurs when w_s <= 2w_n

Nyquist-Shannon Sampling Theorem

if a signal x(t), is band-limited with X(w) = 0 if |w| > w_n , then x(t) is uniquely determined by its samples, x(n) = x(nT) provided that w_s >=2w_n

What’s the Nyquist Rate and Nyquist frequency

2w_n ,The minimum sampling frequency needed to avoid aliasing and f_n is the maximum frequency that is resolved when sampling at the Nyquist rate

For signal which are not low pass signals, How is aliasing avoided

The bandwidth of the signal is used as the Nyquist Frequency.

f_s >= 2B given that B = f_max - f_min

Anti-aliasing filter is needed when

prior to sampling and when the sampling Rate is fixed

Anti-aliasing filter

its an analogue lowpass filter with cut-off frequency = fn (Nyquist Frequency). Bandlimits the analogue input signal to the Nyquist Frequency → fs/2 = fn.

Application of Anti-Aliasing Filter

Digital Camera

The resolution of a camera is fixed and the sampling frequency of an imaging sensor is dependent on the resolution of the imaging sensor. The number of pixels that can be captured.

The bandwidth of the input analogue signal can vary. Many camera use an Optical Low Pass Filter to reduce bandlimit the imaged scene below the Nyquist frequency,

Differences between a Lower Resolution Filter and a Higher Resolution Filter in Imaging

There is overlapping at higher frequencies where the changes in colour occur at a lower pixel rate

Moiré effect

is a visual perception that occurs when viewing a set of lines or dots that is superimposed on another set of lines or dots, where the sets differ in relative size, angle, or spacing

Aliasing Artifact

A pattern that occurs when aliasing occurs showing high frequencies overlapping lower frequencies

Relationship between Continuous time and Discrete time signals

Fundamental Frequency range where the frequency of the CT signal when sampled at a rate fs must be within the range -fs/2 <f<

The highest rate of oscilation of a Discrete time sinusoid is attained

when W = pi and when F = 1/2

What is the Fundamental frequency range of F and f

-fs/2<f<fs/2

-1/2<F< f

The normalised frequency is also known as

Digital frequency or Discrete Time Frequency

The relationship between the digital frequency and the analogue frequency

F = f/fs where fs is the sampling frequency

Ideal reconstruction in the time domain is done by

Passing the digital output signal through an interpolator. The interpolator estimate values of continuous time signal for an intermediate values of time

Ideal reconstruction in the frequency domain

Passing the spectrum through a band limiting filter with its cut of frequency the same as the fundamental frequency of the

The representation of Bandlimiting filter in the time domain is

Sinc function

Mathematically in the frequency domain, Ideal reconstruction is done by

multiplying the spectrum of the ideal reconstruction filter with the spectrum of the sampled sign

Mathematically in the time domain, Ideal reconstruction is done by

convolving the sinc function and a train of impulses (the sampled signal)

In real reconstruction what occurs

The sinc function has an infinite length and therefore other discrete time points are affected

Instead of ideal reconstruction what is used

Sample and hold - This involves holding the impulses in the time domain for a particular duration ensuring that consecutive discrete points are not affected.

They are convolved in the time domain with the sinc function and in frequency domain multiplies by the lowpass filter

Low pass filter in time domain

Smoothens edges of sample & hold operation

Low pass filter in freq domain

Attenuates high frequencies let through by sample & hold filter in frequency domain

The time when sampling ___ by a factor

Normalised by a factor T

Copies of the spectrum are centered at

Sampling frequency

The cut-off frequency of the low pass filter used in reconstruction is half the sampling frequency

True or false

True