EC 2314 Digital Signal Processing By Dr K

EC 2314 Digital Signal Processing By Dr. K. Udhayakumar

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.

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

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

Signal Types • Continuous time – Continuous amplitude • Continuous time – Discrete amplitude • Discrete time – Continuous amplitude • Discrete time – Discrete amplitude

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

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

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

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

A piecewise continuous-time signal • A piecewise continuous-time signal

Discrete Time (DT) Signals • Some real world and many digital signals are discrete time, as they are sampled • E. g. pixels, daily stock price x[n] (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) n

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

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

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+2 ) = cos(t) – sin(t+2 ) = sin(t) – Are both periodic with period 2 2 p

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 • Examples: 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.

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 16/25

Periodic Complex Exponential & Sinusoidal Signals • Consider when a is purely imaginary: • By Euler’s relationship, this can be expressed as: cos(1) • This is a periodic signals because: • when T=2 p/w 0 • A closely related signal is the sinusoidal signal: • We can always use: T 0 = 2 p / w 0 =p T 0 is the fundamental time period w 0 is the fundamental frequency

• 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

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 19/25

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.

• 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:

A piecewise discrete-time signal • A piecewise discrete-time signal

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:

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

• 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 1. Finite total energy (and therefore zero average power) 2. Finite average power (and therefore infinite total energy) Signal analysis over infinite time, all depends on the “tails” (limiting behaviour) •

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

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

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

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

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:

Analog and 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

Even and Odd q Even Signals The continuous-time signal /discrete-time signal if it satisfies the condition is an even • 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

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:

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 square root of -1 is the imaginary part;

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.

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.

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

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

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

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

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

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

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

LTI System Example

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

Digital Signal Processing

Ex)

Properties of LTI Systems • Convolution is commutative • Convolution is distributive

Properties of LTI Systems • Cascade connection of LTI systems

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

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

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].

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

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

• Frequency-Domain Representation

Ex)

Ex)

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 Eigenfunction • is a complex function of frequency Eigenvalue

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

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

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 zero at each value of as does. may not approach but the total energy in the error

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

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

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

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: § DT Systems: (unit pulse) § CT Systems: (impulse) § 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.

Representation of DT Signals Using Unit Pulses

Response of a DT LTI Systems – Convolution DT LTI • 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: convolution operator convolution sum § 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.

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

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

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?

Graphical Convolution 1 2 1 -1 -1 -1 k = -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

Graphical Convolution (Cont. ) 1 2 1 -1 -1 -1 k = -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

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.

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

Properties of Convolution • Commutative: • Distributive: • Associative: • Implications

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

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

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

Exploiting Time-Invariance and Linearity

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]

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

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) signals • Then, their convolution is expressed by the two-sided series

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

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

Sliding over

Sliding over - Cont’d

Plot of

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

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

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

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

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

Frequency domain of downsampling n n n Therefore, the downsampling can be treated as a ‘resampling’ 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

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

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

Downsampling with prefiltering to avoid aliasing (decimation) n n n 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 w. N< /M, where w. N 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.

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

Example of up-sampling Upsampling in the frequency domain

Up-sampling with post low-pass filtering n n 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.

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

Interpolation n Similar to the ideal D/C converter, n n 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]

Sample and hold

Example of sample and hold

Quantizer (Quantization) n n 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. n Typically, we apply two’s complement code for representation.

Quantizer (Quantization)

Quantizer (Quantization) n In general, if we have a (B+1)-bit binary two’s complement fraction of the form: then its value is n n 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

Example of quantization

Analysis of quantization errors n n Quantization error In general, for a (B+1)-bit quantizer with step size , the quantization error satisfies that when n 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.

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

Assumptions about e[n] n n 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. n 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.

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

Example of quantization error 8 -bit quantization error n 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.

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

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

Quantization error analysis n n 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 (d. B), the SNR of a (B+1)-bit quantizer is Hence, the SNR increases approximately 6 d. B for each bit added to the world length of the quantized samples.

Quantization error analysis n n n The equation can be further simplified for analysis. For example, if the signal amplitude has a Gaussian distribution, only 0. 064 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 90 -96 d. B in high-quality music recording and playback requires 16 -bit quantization. n 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.

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 discrete-time equivalent g[n] 119

Spectral Analysis • To ensure bandlimited nature is initially filtered using an analogue antialiasing 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

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

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

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 Rpoint FFT 123

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

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.

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.

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) = 0. 866 x(2) = 0. 866 x(3)= 0 x(4) = -0. 866 x(5) = -0. 866 x(6) = 0

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.

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

Spectral Analysis of Sinusoidal Signals is a periodic function of w with a period 2 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 finitelength version of the sequence given by • 130

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)=2 k, k=0, 1, 2, . . , 63 • One of these points is at given signal frequency of 10 Hz 131

Spectral Analysis of Sinusoidal Signals • DFT magnitude plot 132

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

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

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

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

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

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

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

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

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

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

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

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 2 • Display of is the spectrogram • Display of spectrogram requires normally three dimensions • 144

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

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

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

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

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 149

DSP: applications • Speech, audio – Noise reduction (Dolby), compression (MP 3), … • Radar – filtering, movement detection, … • Image processing – Compression, pattern recognition, segmentation, … • Biomedical – Monitoring, analysis, tele-medicine, …