1. EC 2314 Digital Signal ProcessingBy
Dr. K. Udhayakumar

2. Signal A signal is a pattern of variation that carry information.
Signals are represented mathematically as a function of one or more independent variable
A picture is brightness as a function of two spatial variables, x and y.
In this course signals involving a single independent variable, generally refer to as a time, t are considered. Although it may not represent time in specific application
A signal is a real-valued or scalar-valued function of an independent variable t.

3. Signal Types Signals Continuous-time Discrete-time Continuous-value
Discrete-value
Analog
Digital
Discrete

4. Signal Types Analog signals: continuous in time and amplitude
Example: voltage, current, temperature,…
Digital signals: discrete both in time and amplitude
Example: attendance of this class, digitizes analog signals,…
Discrete-time signals: discrete in time, continuous in amplitude
Example: hourly change of temperature
Theory of digital signals would be too complicated
Requires inclusion of nonlinearities into theory
Theory is based on discrete-time continuous-amplitude signals
Most convenient to develop theory
Good enough approximation to practice with some care
In practice we mostly process digital signals on processors
Need to take into account finite precision effects

5. Signal Types Continuous time – Continuous amplitude Discrete amplitude
Discrete time –

6. Example of signals
Electrical signals like voltages, current and EM field in circuit
Acoustic signals like audio or speech signals (analog or digital)
Video signals like intensity variation in an image
Biological signal like sequence of bases in gene
Noise which will be treated as unwanted signal

7. Signal classification
Continuous-time and Discrete-time
Energy and Power
Real and Complex
Periodic and Non-periodic
Analog and Digital
Even and Odd
Deterministic and Random

8. A continuous-time signal
Continuous-time signal x(t), the independent variable, t is Continuous-time. The signal itself needs not to be continuous.

9. Continuous Time (CT) Signals
Most signals in the real world are continuous time, as the scale is infinitesimally fine.
E.g. voltage, velocity,
Denote by x(t), where the time interval may be bounded (finite) or infinite
x(t) t

10. A piecewise continuous-time signal

11. Discrete Time (DT) Signals
Some real world and many digital signals are discrete time, as they are sampled
E.g. pixels, daily stock price (anything that a digital computer processes)
Denote by x[n], where n is an integer value that varies discretely
Sampled continuous signal
x[n] =x(nk)
x[n] n

12. A discrete-time signal
A discrete signal is defined only at discrete instances. Thus, the independent variable has discrete values only.

13. Sampling
A discrete signal can be derived from a continuous-time signal by sampling it at a uniform rate.
If denotes the sampling period and denotes an integer that may assume positive and negative values,
Sampling a continuous-time signal x(t) at time yields a sample of value
For convenience, a discrete-time signal is represented by a sequence of numbers:
We write
Such a sequence of numbers is referred to as a time series.

14. Periodic Signals
An important class of signals is the class of periodic signals. A periodic signal is a continuous time signal x(t), that has the property
where T>0, for all t.
Examples:
cos(t+2p) = cos(t)
sin(t+2p) = sin(t)
Are both periodic with period 2p 2p

15. Odd and Even Signals
An even signal is identical to its time reversed signal, i.e. it can be reflected in the origin and is equal to the original:
Examples:
x(t) = cos(t)
x(t) = c
An odd signal is identical to its negated, time reversed signal, i.e. it is equal to the negative reflected signal
x(t) = sin(t)
x(t) = t
This is important because any signal can be expressed as the sum of an odd signal and an even signal.

16. Exponential and Sinusoidal Signals
Exponential and sinusoidal signals are characteristic of real-world signals and also from a basis (a building block) for other signals.
A generic complex exponential signal is of the form:
where C and a are, in general, complex numbers. Lets investigate some special cases of this signal
Real exponential signals
Exponential growth
Exponential decay

17. Periodic Complex Exponential & Sinusoidal Signals
Consider when a is purely imaginary:
By Euler’s relationship, this can be expressed as:
This is a periodic signals because:
when T=2p/w0
A closely related signal is the sinusoidal signal:
We can always use:
cos(1)
T0 = 2p/w0
= p
T0 is the fundamental time period
w0 is the fundamental frequency

18. Exponential & Sinusoidal Signal Properties
Periodic signals, in particular complex periodic and sinusoidal signals, have infinite total energy but finite average power.
Consider energy over one period:
Therefore:
Average power:
Useful to consider harmonic signals
Terminology is consistent with its use in music, where each frequency is an integer multiple of a fundamental frequency

19. General Complex Exponential Signals
So far, considered the real and periodic complex exponential
Now consider when C can be complex. Let us express C is polar form and a in rectangular form: So
Using Euler’s relation
These are damped sinusoids

20. Discrete Unit Impulse and Step Signals
The discrete unit impulse signal is defined:
Useful as a basis for analyzing other signals
The discrete unit step signal is defined:
Note that the unit impulse is the first difference (derivative) of the step signal
Similarly, the unit step is the running sum (integral) of the unit impulse.

21. Continuous Unit Impulse and Step Signals
The continuous unit impulse signal is defined:
Note that it is discontinuous at t=0
The arrow is used to denote area, rather than actual value
Again, useful for an infinite basis
The continuous unit step signal is defined:

22. A piecewise discrete-time signal

23. Energy and Power Signals
X(t) is a continuous power signal if:
X[n] is a discrete power signal if:
X(t) is a continuous energy signal if:
X[n] is a discrete energy signal if:

24. Power and Energy in a Physical System
The instantaneous power
The total energy
The average power

25. Energy and Power over Infinite Time
For many signals, we’re interested in examining the power and energy over an infinite time interval (-∞, ∞). These quantities are therefore defined by:
If the sums or integrals do not converge, the energy of such a signal is infinite
Two important (sub)classes of signals
Finite total energy (and therefore zero average power)
Finite average power (and therefore infinite total energy)
Signal analysis over infinite time, all depends on the “tails” (limiting behaviour)

26. Power and Energy
By definition, the total energy over the time interval in a continuous-time signal is:
denote the magnitude of the (possibly complex) number
The time average power
By definition, the total energy over the time interval in a discrete-time signal is:

27. Power and Energy Example 1:
The signal is given below is energy or power signal.
Explain.
This signal is energy signal

28. Power and Energy Example 2:
The signal is given below is energy or power signal.
Explain.
This signal is energy signal

29. Real and Complex A value of a complex signal is a complex number
The complex conjugate, of the signal is;
Magnitude or absolute value
Phase or angle

30. Periodic and Non-periodic
A signal or is a periodic signal if
Here, and are fundamental period, which is the smallest positive values when
Example:

31. Analog and Digital
Digital signal is discrete-time signal whose values belong to a defined set of real numbers
Binary signal is digital signal whose values are 1 or 0
Analog signal is a non-digital signal

32. Even and Odd Even Signals
The continuous-time signal /discrete-time signal is an even signal if it satisfies the condition
Even signals are symmetric about the vertical axis
Odd Signals
The signal is said to be an odd signal if it satisfies the condition
Odd signals are anti-symmetric (asymmetric) about the time origin

33. Even and Odd signals:Facts
Product of 2 even or 2 odd signals is an even signal
Product of an even and an odd signal is an odd signal
Any signal (continuous and discrete) can be expressed as sum of an even and an odd signal:

34. Complex-Valued Signal Symmetry
For a complex-valued signal
is said to be conjugate symmetric if it satisfies the condition
where
is the real part and is the imaginary part;
is the square root of -1

35. Deterministic and Random signal
A signal is deterministic whose future values can be predicted accurately.
Example:
A signal is random whose future values can NOT be predicted with complete accuracy
Random signals whose future values can be statistically determined based on the past values are correlated signals.
Random signals whose future values can NOT be statistically determined from past values are uncorrelated signals and are more random than correlated signals.

36. Deterministic and Random signal(contd…)
Two ways to describe the randomness of the signal are:
Entropy:
This is the natural meaning and mostly used in system performance measurement.
Correlation:
This is useful in signal processing by directly using correlation functions.

37. Basic sequences and sequence operations
Delaying (Shifting) a sequence
Unit sample (impulse) sequence
Unit step sequence
Exponential sequences

38. Discrete-Time Systems
A Discrete-Time System is a mathematical operation that maps a given input sequence x[n] into an output sequence y[n]
Example:
Moving (Running) Average
Maximum
Ideal Delay System

39. Memoryless System
A system is memoryless if the output y[n] at every value of n depends only on the input x[n] at the same value of n
Example :
Square
Sign
counter example:
Ideal Delay System

40. Linear Systems Linear System: A system is linear if and only if
Example: Ideal Delay System

41. Time-Invariant Systems
Time-Invariant (shift-invariant) Systems
A time shift at the input causes corresponding time-shift at output
Example: Square
Counter Example: Compressor System

42. Causal System
A system is causal iff it’s output is a function of only the current and previous samples
Examples: Backward Difference
Counter Example: Forward Difference

43. Stable System
Stability (in the sense of bounded-input bounded-output BIBO). A system is stable iff every bounded input produces a bounded output
Example: Square
Counter Example: Log

44. LTI System Example

45. Linear Time-Invariant Systems
Special importance for their mathematical tractability
Most signal processing applications involve LTI systems
LTI system can be completely characterized by their impulse response

46. Digital Signal Processing

47. Ex)

49. Properties of LTI Systems
Convolution is commutative
Convolution is distributive

50. Properties of LTI Systems
Cascade connection of LTI systems

51. Stable and Causal LTI Systems
An LTI system is (BIBO) stable iff Impulse response is absolute summable
Let’s write the output of the system as
Then the output is bounded by
The output is bounded if the absolute sum is finite
An LTI system is causal iff

52. Stability Condition : A linear time-invariant system is stable If and only if

53. Causality Condition :
Neither necessary nor sufficient
condition for all systems,
But necessary and sufficient for LTI system
But x[n-k] for k>=0 shows
The future values of x[n].
So y[n] depends only on the
Future values of x[n].

54. Linear Constant-Coefficient Difference Equations
An important class of LTI systems of the form
The output is not uniquely specified for a given input
The initial conditions are required
Linearity, time invariance, and causality depend on the initial conditions
If initial conditions are assumed to be zero system is linear, time invariant, and causal
Example
Moving Average

55. Linear Constant-Coefficient Difference Equations
Ex)
X[n] is the difference of y[n]

56. Frequency-Domain Representation

57. Ex)

59. Ex)

60. Eigenfunctions of LTI Systems
Complex exponentials are eigenfunctions of LTI systems:
Let’s see what happens if we feed x[n] into an LTI system:
The eigenvalue is called the frequency response of the system
is a complex function of frequency
Eigenfunction
Eigenvalue

61. Discrete-Time Fourier Transform
Many sequences can be expressed as a weighted sum of complex exponentials as
Where the weighting is determined as
is the Fourier spectrum of the sequence x[n]
The phase wraps at 2 hence is not uniquely specified
The frequency response of a LTI system is the DTFT of the impulse response

62. Absolute and Square Summability
For a given sequence if the infinite sum convergence, the DTFT exist
All stable systems are absolute summable and have finite and continues frequency response

63. Absolute and Square Summability
Absolute summability is sufficient condition for DTFT
Some sequences may not be absolute summable but only square summable
Such sequences can be represented by fourier transform if
In other words, the error may not approach zero at each value of as but the total energy in the error does.

64. Example: Ideal Lowpass Filter
The periodic DTFT of the ideal lowpass filter is
The inverse can be written as
Not causal, Not absolute summable but it has a DTFT, The DTFT converges in the mean-squared sense
Role of Gibbs phenomenon

65. Ex)
The impulse response is not causal,
Not absolutely summable, but squarely summable,
Since sequence values approach zero as n-> infinity,
But only as 1/n

66. Symmetric Sequence and Functions
Conjugate-symmetric
Conjugate-antisymmetric
Sequence
Function

67. Exploiting Superposition and Time-Invariance
DT LTI
System
Are there sets of “basic” signaxk[n], such that:
We can represent any signal as a linear combination (e.g, weighted sum) of these building blocks? (Hint: Recall Fourier Series.)
The response of an LTI system to these basic signals is easy to compute and provides significant insight.
For LTI Systems (CT or DT) there are two natural choices for these building blocks:
Later we will learn that there are many families of such functions: sinusoids, exponentials, and even data-dependent functions. The latter are extremely useful in compression and pattern recognition applications.
DT Systems: (unit pulse)
CT Systems: (impulse)

68. Representation of DT Signals Using Unit Pulses

69. Response of a DT LTI Systems – Convolution
Define the unit pulse response, h[n], as the response of a DT LTI system to a unit pulse function, [n].
Using the principle of time-invariance:
Using the principle of linearity:
Comments:
Recall that linearity implies the weighted sum of input signals will produce a similar weighted sum of output signals.
Each unit pulse function, [n-k], produces a corresponding time-delayed version of the system impulse response function (h[n-k]).
The summation is referred to as the convolution sum.
The symbol “*” is used to denote the convolution operation.
convolution operator
convolution sum

70. LTI Systems and Impulse Response
The output of any DT LTI is a convolution of the input signal with the unit pulse response:
Any DT LTI system is completely characterized by its unit pulse response.
Convolution has a simple graphical interpretation:
DT LTI

71. Visualizing Convolution
There are four basic steps to the calculation:
The operation has a simple graphical interpretation:

72. Calculating Successive Values
We can calculate each output point by shifting the unit pulse response one sample at a time:
y[n] = 0 for n < ???
y[-1] =
y[0] =
y[1] = …
y[n] = 0 for n > ???
Can we generalize this result?

73. Graphical Convolution2 1 -1 -1 1 -1
k =

74. Graphical Convolution (Cont.)2 1 -1 -1 1 -1
k =

75. Graphical Convolution (Cont.)
Observations:
y[n] = 0 for n > 4
If we define the duration of h[n] as the difference in time from the first nonzero sample to the last nonzero sample, the duration of h[n], Lh, is 4 samples.
Similarly, Lx = 3.
The duration of y[n] is: Ly = Lx + Lh – 1. This is a good sanity check.
The fact that the output has a duration longer than the input indicates that convolution often acts like a low pass filter and smoothes the signal.

76. Examples of DT Convolution
Example: unit-pulse
Example: delayed unit-pulse
Example: unit step
Example: integration

77. Properties of Convolution
Commutative:
Implications
Distributive:
Associative:

78. Useful Properties of (DT) LTI Systems
Causality:
Stability:
Bounded Input ↔ Bounded Output
Sufficient Condition:
Necessary Condition:

79. Convolution Representation..Example
Consider the DT system described by
Its impulse response can be found to be

80. Representing Signals in Terms of Shifted and Scaled Impulses
Let x[n] be an arbitrary input signal to a DT LTI system
Suppose that for
This signal can be represented as

81. Exploiting Time-Invariance and Linearity

82. The Convolution Sum
This particular summation is called the convolution sum
Equation is called the convolution representation of the system
Remark: a DT LTI system is completely described by its impulse response h[n]

83. Block Diagram Representation of DT LTI Systems
Since the impulse response h[n] provides the complete description of a DT LTI system, we write

84. The Convolution Sum for Noncausal Signals
Suppose that we have two signals x[n] and v[n] that are not zero for negative times (noncausal signals)
Then, their convolution is expressed by the two-sided series

85. Example: Convolution of Two Rectangular Pulses
Suppose that both x[n] and v[n] are equal to the rectangular pulse p[n] (causal signal) depicted below

86. The Folded Pulse
The signal is equal to the pulse p[i] folded about the vertical axis

87. Sliding over

88. Sliding over Cont’d

89. Plot of

90. Properties of the Convolution Sum
Associativity
Commutativity
Distributivity w.r.t. addition

91. Properties of the Convolution Sum - Cont’d
Shift property: define
Convolution with the unit impulse
Convolution with the shifted unit impulse
then

92. Changing the Sampling rate using discrete-time processing
downsampling; sampling rate compressor;

93. Frequency domain of downsampling
Since this is a ‘re-sampling’ process. Remember that, from continuous-time sampling of x[n]=xc(nT), we have
Similarly, for the down-sampled signal xd[m]=xc(mT’), (where T’ = MT), we have

94. Frequency domain of downsampling
We are interested in the relation between X(ejw) and Xd(ejw). Let’s represent r as r = i + kM, where 0  i  M1, (i.e., r  i (mod M)). Then

95. Frequency domain of downsampling
Therefore, the downsampling can be treated as a ‘re-sampling’ process. It s frequency domain relationship is similar to that of the D/C converter as:
This is equivalent to compositing M copies of the of X(ejw), frequency scaled by M and shifted by inter multiples of 2.
The aliasing can be avoided by ensuring that X(ejw) is bandlimited as

96. Example of downsampling in the Frequency domain (without aliasing)
Sampling with a sufficiently large rate which avoids aliasing

97. Example of downsampling in the Frequency domain (without aliasing)
Downsampling by 2 (M=2)

98. Downsampling with prefiltering to avoid aliasing (decimation)
From the above, the DTFT of the down-sampled signal is the superposition of M shifted/scaled versions of the DTFT of the original signal.
To avoid aliasing, we need wN</M, where wN is the highest frequency of the discrete-time signal x[n].
Hence, downsampling is usually accompanied with a pre-low-pass filtering, and a low-pass filter followed by down-sampling is usually called a decimator, and termed the process as decimation.

99. Up-sampling : Upsampling; sampling rate expander. or equivalently,
In frequency domain: :

100. Example of up-sampling
Upsampling in the frequency domain

101. Up-sampling with post low-pass filtering
Similar to the case of D/C converter, upsampoling is usually companied with a post low-pass filter with cutoff frequency /L and gain L, to reconstruct the sequence.
A low-pass filter followed by up-sampling is called an interpolator, and the whole process is called interpolation.

102. Example of up-sampling followed by low-pass filtering
Applying low-pass filtering

103. Interpolation Similar to the ideal D/C converter,
If we choose an ideal lowpass filter with cutoff frequency /L and gain L, its impulse response is
Hence
Its an interpolation of the discrete sequence x[k]

104. Sample and hold

105. Example of sample and hold

106. Quantizer (Quantization)
The real-valued signal has to be stored as a code for digital processing. This step is called quantization.
The quantizer is a nonlinear system.
Typically, we apply two’s complement code for representation.

107. Quantizer (Quantization)

108. Quantizer (Quantization)
In general, if we have a (B+1)-bit binary two’s complement fraction of the form:
then its value is
The step size of the quantizer is
where Xm is the full scale level of the A/D converter.
The numerical relationship beween the code words and the quantizer samples is

109. Example of quantization

110. Analysis of quantization errors
In general, for a (B+1)-bit quantizer with step size , the quantization error satisfies that
when
If x[n] is outside this range, then the quantization error is larger in magnitude than /2, and such samples are saided to be clipped.

111. Analysis of quantization errors
Analyzing the quantization by introducing an error source and linearizing the system:
The model is equivalent to quantizer if we know e[n].

112. Assumptions about e[n]
e[n] is a sample sequence of a stationary random process.
e[n] is uncorrelated with the sequence x[n].
The random variables of the error process e[n] are uncorrelated; i.e., the error is a white-noise process.
The probability distribution of the error process is uniform over the range of quantization error (i.e., without being clipped).
The assumptions would not be justified. However, when the signal is a complicated signal (such as speech or music), the assumptions are more realistic.
Experiments have shown that, as the signal becomes more complicated, the measured correlation between the signal and the quantization error decreases, and the error also becomes uncorrelated.

113. Example of quantization error
original signal
3-bit quantization result
3-bit quantization error

114. Example of quantization error
8-bit quantization error
In a heuristic sense, the assumptions of the statistical model appear to be valid if the signal is sufficiently complex and the quantization steps are sufficiently small, so that the amplitude of the signal is likely to traverse many quantization steps from sample to sample.

115. Quantization error analysis
e[n] is a white noise sequence. The probability density function of e[n] is

116. Quantization error analysis
The mean value of e[n] is zero, and its variance is
Since
For a (B+1)-bit quantizer with full-scale value Xm, the noise variance, or power, is

117. Quantization error analysis
A common measure of the amount of degradation of a signal by additive noise is the signal-to-noise ratio (SNR), defined as the ratio of signal variance (power) to noise variance. Expressed in decibels (dB), the SNR of a (B+1)-bit quantizer is
Hence, the SNR increases approximately 6dB for each bit added to the world length of the quantized samples.

118. Quantization error analysis
The equation can be further simplified for analysis. For example, if the signal amplitude has a Gaussian distribution, only percent of the samples would have an amplitude greater than 4x.
Thus to avoid clipping the peaks of the signal (as is assumed in our statistical model), we might set the gain of filters and amplifiers preceding the A/D converter so that x = Xm/4. Using this value of x gives
For example, obtaining a SNR about dB in high-quality music recording and playback requires 16-bit quantization.
But it should be remembered that such performance is obtained only if the input signal is carefully matched to the full-scale of the A/D converter.

119. Spectral Analysis
Spectral analysis is concerned with the determination of the energy or power spectrum of a continuous-time signal
It is assumed that is sufficiently bandlimited so that its spectral characteristics are reasonably estimated from those of its of its discrete-time equivalent g[n]
119

120. Spectral Analysis
To ensure bandlimited nature is initially filtered using an analogue anti-aliasing filter the output of which is sampled to provide g[n]
Assumptions:
(1) Effect of aliasing can be ignored
(2) A/D conversion noise can be neglected
120

121. Spectral Analysis Three typical areas of spectral analysis are:
1) Spectral analysis of stationary sinusoidal signals
2) Spectral analysis of of nonstationary signals
3) Spectral analysis of random signals
121

122. Spectral Analysis of Sinusoidal Signals
Assumption - Parameters characterising sinusoidal signals, such as amplitude, frequency, and phase, do not change with time
For such a signal g[n], the Fourier analysis can be carried out by computing the DTFT
122

123. Spectral Analysis of Sinusoidal Signals
Initially the infinite-length sequence g[n] is windowed by a length-N window w[n] to yield
DTFT of then is assumed to provide a reasonable estimate of
is evaluated at a set of R ( ) discrete angular frequencies using an R-point FFT
123

124. Spectral Analysis of Sinusoidal Signals
Note that
The normalised discrete-time angular frequency corresponding to DFT bin k is
while the equivalent continuous-time angular frequency is
124

125. Sampling and Aliasing..Overview
Periodic sampling, the process of representing a continuous signal with a sequence of discrete data values, pervades the field of digital signal processing.
In practice, sampling is performed by applying a continuous signal to an analog-to-digital (A/D) converter whose output is a series of digital values.

126. Cont..
With regard to sampling, the primary concern is how fast must the given continuous signal be sampled in order to preserve its information content.

127. ALIASING
There is a frequency-domain ambiguity associated with the discrete-time signal samples that is absent in the continuous signal world.
eg. Suppose you are given the following sequence of values,
x(0) = 0 x(1) = x(2) = x(3)= 0
x(4) = x(5) = x(6) = 0

128. Oversampling
If the original waveform does not vary much over the duration of p(t), then we will also obtain a good construction. Oversampling, i.e., using a sampling rate that is much greater than the Nyquist rate, can ensure this.

129. Spectral Analysis of Sinusoidal Signals
Consider
expressed as
Its DTFT is given by
129

130. Spectral Analysis of Sinusoidal Signals
is a periodic function of w with a period 2p containing two impulses in each period
In the range , there is an impulse at
of complex amplitude and an impulse at of complex amplitude
To analyze g[n] using DFT, we employ a finite-length version of the sequence given by
130

131. Spectral Analysis of Sinusoidal Signals
Example - Determine the 32-point DFT of a length-32 sequence g[n] obtained by sampling at a rate of 64 Hz a sinusoidal signal of frequency 10 Hz
Since Hz the DFT bins will be located in Hz at ( k/NT)=2k, k=0,1,2,..,63
One of these points is at given signal frequency of 10Hz
131

132. Spectral Analysis of Sinusoidal Signals
DFT magnitude plot
132

133. Spectral Analysis of Sinusoidal Signals
Example - Determine the 32-point DFT of a length-32 sequence g[n] obtained by sampling at a rate of 64 Hz a sinusoid of frequency 11 Hz
Since
the impulse at f = 11 Hz of the DTFT appear between the DFT bin locations k = 5 and k = 6
the impulse at f= -11 Hz appears between the DFT bin locations k = 26 and k = 27
133

134. Spectral Analysis of Sinusoidal Signals
DFT magnitude plot
Note: Spectrum contains frequency components at all bins, with two strong components at k = 5 and k = 6, and two strong components at k = 26 and k = 27
134

135. Spectral Analysis of Sinusoidal Signals
The phenomenon of the spread of energy from a single frequency to many DFT frequency locations is called leakage
Problem gets more complicated if the signal contains more than one sinusoid
135

136. Spectral Analysis of Sinusoidal Signals
Example -
From plot it is difficult to determine if there is one or more sinusoids in x[n] and the exact locations of the sinusoids
136

137. Spectral Analysis of Sinusoidal Signals
An increase in resolution and accuracy of the peak locations is obtained by increasing DFT length to R = 128 with peaks occurring at k = 27 and k =45
137

138. Spectral Analysis of Sinusoidal Signals
Reduced resolution occurs when the difference between the two frequencies becomes less than 0.4
As the difference between the two frequencies gets smaller, the main lobes of the individual DTFTs get closer and eventually overlap
138

139. Spectral Analysis of Nonstationary Signals
An example of a time-varying signal is the chirp signal and shown below for
The instantaneous frequency of x[n] is
139

140. Spectral Analysis of Nonstationary Signals
Other examples of such nonstationary signals are speech, radar and sonar signals
DFT of the complete signal will provide misleading results
A practical approach would be to segment the signal into a set of subsequences of short length with each subsequence centered at uniform intervals of time and compute DFTs of each subsequence
140

141. Spectral Analysis of Nonstationary Signals
The frequency-domain description of the long sequence is then given by a set of short-length DFTs, i.e. a time-dependent DFT
To represent a nonstationary x[n] in terms of a set of short-length subsequences, x[n] is multiplied by a window w[n] that is stationary with respect to time and move x[n] through the window
141

142. Spectral Analysis of Nonstationary Signals
Four segments of the chirp signal as seen through a stationary length-200 rectangular window
142

143. Short-Time Fourier Transform
Short-time Fourier transform (STFT), also known as time-dependent Fourier transform of a signal x[n] is defined by
where w[n] is a suitably chosen window sequence
If w[n] = 1, definition of STFT reduces to that of DTFT of x[n]
143

144. Short-Time Fourier Transform
is a function of 2 variables: integer time index n and continuous frequency w
is a periodic function of w with a period 2p
Display of is the spectrogram
Display of spectrogram requires normally three dimensions
144

145. Short-Time Fourier Transform
Often, STFT magnitude is plotted in two dimensions with the magnitude represented by the intensity of the plot
Plot of STFT magnitude of chirp sequence
with for a length of 20,000 samples computed using a Hamming window of length 200 shown next
145

146. Short-Time Fourier Transform
STFT for a given value of n is essentially the DFT of a segment of an almost sinusoidal sequence
146

147. Short-Time Fourier Transform
Shape of the DFT of such a sequence is similar to that shown below
Large nonzero-valued DFT samples around the frequency of the sinusoid
Smaller nonzero-valued DFT samples at other frequency points
147

148. STFT on Speech
An example of a narrowband spectrogram of a segment of speech signal
148

149. STFT on Speech
The wideband spectrogram of the speech signal is shown below
The frequency and time resolution tradeoff between the two spectrograms can be seen
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150. DSP: applications Speech, audio Radar Image processing Biomedical
Noise reduction (Dolby), compression (MP3), …
Radar
filtering, movement detection, …
Image processing
Compression, pattern recognition, segmentation,…
Biomedical
Monitoring, analysis, tele-medicine, …