Digital Signal Processing Discussion #3 Signals Operations Tarun arun Choubi Choubisa sa Dept. of ETC, KIIT University
18 J
2011
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Classification: Causal/non-causal/an Causal/non-causal /anti-causal ti-causal •
A system for which the output at any instant depends only on the past or/and present values of the input( not on future samples) is called as causal system. Referred Referred to as nonanticipative,, as the system output does not anticipate future values of the input anticipative
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E.g. y(n)= n*x(n) , y(n)=x(n) +x(n-1)
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All real-time physical systems are causal causal,, because time only moves forward.
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Causality does not apply to systems processing recorded signals, e.g. taped sports games vs. live broadcast. Fact: –
A causal system may be memory or memory-less memor y-less system.
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Any memoryless memoryless system system is causal.
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The composition of causal systems is causal
A system for which the output at any instant depends also on future values (in addition to possible dependence on past or current input values)of the input , is called as non-causal (acausal) system. A non-causal system is also called a non-realizable system. E.g. y(n)=x(n2 ) , y(n)=x(-n) y(n)=x(-n) , y(n) = x(n/3), y(n)=x(n)+x(n+1) y(n)=x(n)+x(n+1)
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Classification: Causal/non-causal/an Causal/non-causal /anti-causal ti-causal •
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A system that depends solely on future input values is an antic anticausal ausal sy syste stem m. Eg: y(n) = x(n+1), prediction of current value from only future values in the corrupted CD. Fact: All anti-causal /non causal systems are memory systems systems but opposite oppo site is not true. To check always take take negative, 0, positive values and specially -1 < value < 1 Observations: Negative Negative index, index index scaling, and power of index represent represent noncausality.
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Classification: Stable/uns Stable/unstable table •
The system is said to be stable if any bounded(amplitude) input signal results in bounded output signal –
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Unbounded signals r(n) , n*u(n)
E.g. Consider the DT system of the bank account y[ n]
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18 J
Stable system y[ y[n] = ( x[ x[n])2 Suppose x[n] is limited to the ran ange ge -10 -10 < x[ x[n] n] < 10 10? ?
where a>0
The system is said to be unstable if the system gives unbounded output signal in response to bounded input signal –
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bounded signals u(n) , e-an
x[ n]
1.01 y[n 1]
This grows without bound, due to to 1.01 multiplier multiplie r. This system is unstable. 201 1
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Signal Operations •
Time Shifting – –
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Time Reversal: negate the index or time(n=-n). Time Scaling – – –
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Delaying(n=n-k) Advancing(n=n+k)
In Discrete Time it can also term as Rate Changing Sampling rate can be changed to up or down Up sampling/ Down sampling
Amplitude Scaling: Scaling: each sample of the signal would be scaled by scaling Addition/Subtraction:: corresponding samples from both signals would be Addition/Subtraction added, subtracted Multiplication:: corresponding samples from both signals would be Multiplication multiplied
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Signal Opera Operations: tions: Time Rever Reversing sing (inv (inversion ersion)) To time-reverse a signal, replace every t with –t. So, x(-t) represents the time reversal (or inverse) of x(t). The graph graph of x(-t) can can be forme formed d by rotat rotating ing the graph graph of x(t) x(t) 180 about the y-axis(mirror image about y axis). Example
x(t) 10
t
0 x(-t) 10
t 0
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Signal Opera Operations: tions: Time Rever Reversing sing (inv (inversion ersion))
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Signal Opera Operations: tions: Time Rever Reversing sing (inv (inversion ersion)) Example: Given x(t) below, sketch x(-t). x(t) 10
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x(-t)
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Signal Operations: Time Reversing
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Signal Operations: Time Shifting DT
Replacing every n in a waveform wavef orm with n – N shifts the waveform N samples to the right. In general, a negative shift is a shift to the right (delaying). Similarly, a positive shift is a shift shift to the left (advancing).
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Signal Operations: Time Shifting CT
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Signal Operations: Time Shifting CT Example: Given x(t) x( t) below, sketch x1(t) = x(t – 1) and x2(t) = x(t + 1). x(t) 10
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x1(t) = x(t x(t - 1)
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x2(t) = x(t + 1)
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Signal Operations: Time Scaling Time scaling
Example
x(t)
Time scaling is the compression or expansion of a signal.
Original signal
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Compressed signal (t) = x(2t) is a compressed version of x(t) as shown on the right.
T1
T2
0 (t) = x(2t)
Compressed signal (a = 2)
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In general, (t) = x(at) represents a compressed signal if a if a > 1.
T1
Expanded signal
0
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Similarly, (t) = x(at) represents an expanded signal if a if a < 1.
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T2 2
(t) = x(t/2)
Expanded signal (a = 0.5)
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To scale any function by a, replace each t by at in the function.
t
t 2T1
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Signal Operations: Time Scaling
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Signal Operations: Time Scaling Example: Given x(t) below, sketch x 1(t) = x(2t) and x2(t) = x(0.5t) = x(t/2). x(t) 10
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x1(t) = x(2t)
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x2(t) = x(t/2)
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Signal Operations: Time Scaling sketch x( x(tt), x1(t) = x(2t), and x2(t) = x(t/2). Example: If x(t) = 10sin(4 t - ), sk x(t)
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x1(t) = x(2t)
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x2(t) = x(t/2)
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Effect Eff ect of time scaling on frequency: __________________ _________________________________ _______________ Effect Eff ect of time scaling on amplitude: __________________ _________________________________ _______________ 16
Signal Operations: Time Scaling
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TSh, TR, TS
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Order of operations –
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Time shifting(TSh) Time reversal(TR) Time scaling(TS) Amplitude scaling
TSh and TR TR are not commut commutativ ative. e.
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Amplitude Scaling •
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It changes the amplitude of the signal by a scaling factor. Some amplifiers not only amplify signals but also add (or remove) a constant, constant, or dc, value.
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Combined oper operations ations We can use various combinations of the three operations just operations just covered: covered: time shifting, time scaling, and time reversal. reversal. The operations can often be applied in different orders,, but care must be taken. orders Example: To form form x(at - b) from x(t) we could could use two approaches: approaches: 1) Time Time-s -shi hift ft the then n timetime-sc scal ale e A. Time-s Time-shif hiftt x(t) x(t) by b to obta obtain in x(t x(t - b). I.e., I.e., repla replace ce eve every ry t by t - b. B. Time-s Time-scal cale e x(t x(t - b) by by a (i.e. (i.e.,, repla replace ce t by by at) at) to form form x(at x(at - b) 2) Time Time-s -sca cale le the then n timetime-sh shif iftt A. Time-s Time-scal cale e x(t) x(t) by by a to obtain obtain x(at). x(at). B. Time-shi Time-shift ft x(at) by b/a b/a (i.e., (i.e., repla replace ce t with t – b/a) to yield x(a[t – b/a]) = x(at – b) Standard order order is (1) (1) Time Time Shif Shifti ting ng (2) (2) Time Time rev rever ersa sall (3) (3) Time Time scal scalin ing g
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Combined oper operations ations Example: Given x(t) below, sketch x 1(t) = x(2t x(2t - 1) and and x2(t) = x(t/2 + 1). x(t) 10
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x1(t) = x(2t x(2t - 1)
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x2(t) = x(t/2 + 1)
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Signal Addition
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Signal Addition: Saturation of color •
Color tree Moving along a radius of a circle changes the saturation (vividness) of a color(signal addition: white color is added.)
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hue l i g h t n e s s
Moving up the tree increases the lightness of a color Moving around a circle of given radius changes the hue of a color(different frequencies) These three coordinates can be described in terms of three numbers
Photoshop: uses H, S and B
saturation
Acknowledgement •
Various graphics used here has been taken from public resources instead of redrawing it. Thanks to those who have created created it.
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Thanks to:
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Prof. John G. Proakis
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Prof Prof. Dimitris Dimitris G. Manolakis Manolakis
