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
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A Fourier Transforms relationship with Fourier series is
Finding the Fourier series of a periodic signal with its period tending to infinity
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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
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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
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Effect of Pulse width using the Forward FT
The spectrum of it Forward FT shrinks with an increase in the pulse of the signal
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For a constant signal For.FT i.e tao of a pulse → 0
A impulse in the FT
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Shifting a signal in time changes
Linearly the phase of its spectrum and the magnitude is not affected
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real signal spectrum displays Hermitian symmetry t or f
true - symmetric amplitude and odd symmetric phase
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Fourier Transform of periodic signals is the
Fourier series
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The A/D Converter comprises of
Sampler, Quantizer and Coder
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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.
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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
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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
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A phenomenon where instead of original copy of the input when reconstruction occurs, A different spectrum is received.
Aliasing. Occurs when w_s
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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
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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
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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
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Anti-aliasing filter is needed when
prior to sampling and when the sampling Rate is fixed
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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.
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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,
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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
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**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
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Aliasing Artifact
A pattern that occurs when aliasing occurs showing high frequencies overlapping lower frequencies
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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
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The highest rate of oscilation of a Discrete time sinusoid is attained
when W = pi and when F = 1/2
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What is the Fundamental frequency range of F and f
\-fs/2
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The normalised frequency is also known as
Digital frequency or Discrete Time Frequency
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The relationship between the digital frequency and the analogue frequency
F = f/fs where fs is the sampling frequency
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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
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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
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The representation of Bandlimiting filter in the time domain is
Sinc function
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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
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Mathematically in the time domain, Ideal reconstruction is done by
convolving the sinc function and a train of impulses (the sampled signal)
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In real reconstruction what occurs
The sinc function has an infinite length and therefore other discrete time points are affected
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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
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Low pass filter in time domain
Smoothens edges of sample & hold operation
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Low pass filter in freq domain
Attenuates high frequencies let through by sample & hold filter in frequency domain
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The time when sampling ___ by a factor
Normalised by a factor T
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Copies of the spectrum are centered at
Sampling frequency
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The cut-off frequency of the low pass filter used in reconstruction is half the sampling frequency