1. CT-321 Digital Signal Processing
Yash Vasavada
Autumn 2016
DA-IICT
Lecture 4
LSI Systems and DTFT
9th August 2016

2. Review and Preview Review of past lecture:
Frequency Domain Representations
Can simplify complicated-looking time domain representation.
NCO as an implementation of complex-phasor
Rectangular pulse in time domain
Spreading in this pulse in time domain results in a localization of its spectrum in frequency domain, and vice versa
Rectangular pulse for windowing
How to perform frequency domain windowing
Averaging multiple samples of a complex phasor at frequency 𝑓, and its relation to DTFT

3. Preview and Reading Assignment
Preview of this lecture:
Linear Time Invariant (LTI) Systems
Relation to DTFT
Reading Assignment
OS, 3rd Edition: Sections 1.2 to 1.4, and Sections 1.6 to 1.9
Note: available in the library
PM: Sections 2.1 to 2.3

4. Review of Past Lecture: Rectangular Window Function
Rectangular pulse is a key signal of frequent application in theoretical DSP studies
To see why, consider a hypothetical signal that we have received, that has an undesired component mixed in with the desired signal. We want to get rid of the unwanted component.
One solution is to multiply this received signal by a rectangular pulse of unity amplitude and of sufficient width centered at the desired signal’s location.
This will zero out the unwanted component and leave the desired part unchanged.
This operation is also called rectangular windowing
Notice that this mixture of desired and unwanted signals can occur either in time domain or in frequency domain
In time domain, it’s not hard to envision implementation of the multiplication operation between the received signal and the rectangular pulse. However, how to do so in the frequency domain? 1
Unwanted
Desired
Time or Frequency

5. Review of Past Lecture: Averaging of Multiple Samples of Complex Phasor
We have so far looked at the frequency domain representation of the rectangular pulse in continuous time domain. What about discrete time domain?
A useful mathematical series for DSP engineers: a sum of (2𝑁+1) samples of a phasor rotating at frequency 𝐹 from sample index −𝑁 to 𝑁
Example application: comm receiver operating with a frequency offset of F
How to derive a closed-form expression of 𝑆 𝐹 ?  Begin by writing 𝑆(𝐹) as follows:
Closed form expression of 𝑆 𝐹 :
Surprisingly quite similar to the Fourier Transform of the rectangular pulse in continuous time domain!
Why?
= 2𝑁+1 ×𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑎𝑙𝑢𝑒
Now solve using this expression:

6. Discrete Time Fourier Transform
Discrete Time Fourier Transform (DTFT) allows us to obtain frequency domain representation of discrete time signals
Discrete time analogue of the conventional (continuous time) Fourier Transform
Defined as follows:
Several items to note:
DTFT is evaluated for a given value of frequency 𝑓.
What is the range of this frequency 𝑓? Is 𝑓 continuous (analog) or discrete?

7. Recap
We have covered a lot of ground – and the different parts of it looked much different without any obvious interconnections
Looked at the need and the benefit of windowing by a rectangular window
Although the benefit is clear, we are not yet able to answer the question of how to do this windowing in frequency domain
Visited the commonly-invoked process of averaging (elementary school math) when it is applied to a complex phasor
Invoked high-school math on how to sum a finite-length geometric series
Realized that the formula for this average value, which is based on SINC function whose argument is a product of the frequency parameter and the time duration of averaging process, is similar to the Fourier Transform of C-T rectangular function
Defined DTFT of a D-T sequence 𝑥(𝑛)
Note: averaging in item 2 above is just the DTFT of rectangular sequence 𝑥 𝑛 = 𝑟 𝑁 (𝑛)
Need to still answer item # 2

8. DTFT of the Rectangular Pulse
(Frequency Domain Representation of Rectangular Pulses)
Time Domain Rectangular Pulses
Frequency domain representation becomes localized (is narrower) as the time domain signal spreads out (becomes wider)
Same as the observation we have made earlier in context of C-T signals
Can DTFT be reversed?
i.e., can the discrete time samples be recovered from continuous- frequency DTFT?
If so, what is likely to be this “inverse” DTFT of rectangular function in the frequency domain?
𝑁=2
Peak value is given as 2𝑁+1. Why?
Locations of the first nulls are always at 𝑓= ±1 2𝑁+1
𝑁=4
Values at the extreme points of 𝑓=± 1 2 are always the same and they are either +1 or −1 𝑓

9. Inverse DTFT Inverse DTFT is defined as follows:
Consider a periodic rectangular window in frequency domain: Π 𝑓 𝑓 𝑐 =1, 0≤ 𝑓 ≤ 𝑓 𝑐 2
Inverse DTFT of Π 𝑓 𝑓 𝑐 is 𝑓 𝑐 sinc( 𝑓 𝑐 𝑛):
Time Domain: Discrete Samples
Frequency Domain (Continuous and Periodic)
𝑓 𝐶 sinc( 𝑓 𝑐 𝑛)
Π 𝑓 𝑓 𝑐

10. Why DTFT is Important?
Recall our conundrum earlier: how to multiply any signal with another signal (say, a rectangular window function) in frequency domain?
Armed with the tool of DTFT, we are now in a position to begin to solve this puzzle

11. Digital Signal Processing Systems
Let us refer to a digital signal processor that accomplishes this task of frequency domain multiplication (or any such signal processing task) as a system 𝕋
It takes as the input a discrete-time sequence 𝑥(𝑛)
It generates as the output another (transformed) discrete-time signal 𝑦(𝑛)
There are certain number of key characteristics or properties that any such system exhibits. We need to understand and be aware of these
𝑥(𝑛)
y(𝑛) 𝕋

12. Digital Signal Processing Systems: Some Key Properties
Key questions to ask for any system 𝕋, with input 𝑥(𝑛) and output 𝑦 𝑛 , i.e., 𝑦 𝑛 =𝕋 𝑥(𝑛) :
What is the additive property of the system?
When multiple inputs 𝑥 1 (𝑛), 𝑥 2 𝑛 ,…, x M (n) are added together and fed as the input to the system 𝕋…
…Is the output the same as the case in which one feeds individual 𝑥 𝑚 (𝑛) to the system, obtains the individual output 𝑦 𝑚 𝑛 , and adds them up?
For systems satisfying the additive (or superposition) property: 𝕋 𝑚=1 𝑀 𝑥 𝑚 (𝑛) = 𝑚=1 𝑀 𝕋 𝑥 𝑚 (𝑛)
What is the multiplicative or scaling property of the system?
If input 𝑥(𝑛) is multiplied by a constant, what happens to the output?
For certain systems 𝕋, output is multiplied by the same constant; i.e., 𝕋 𝑎𝑥(𝑛) =𝑎𝕋 𝑥(𝑛) =𝑎𝑦 𝑛
Amplitude of the output is proportional to the amplitude of the input
However, for the others, the output cannot be so expressed; i.e.,𝕋 𝑎𝑥(𝑛) ≠𝑎𝕋 𝑥(𝑛)
Amplitude proportionality does not hold
Systems that satisfy both the additive and multiplicative properties are called linear systems

13. Digital Signal Processing Systems: Some Key Properties
Consider the two systems A and B shown below 𝑎 𝑎
𝑥 1 (𝑛)
𝑥 1 (𝑛) 𝕋
𝑦 𝐴 (𝑛)
𝑦 𝐵 (𝑛) 𝕋
𝑥 2 (𝑛)
𝑥 2 (𝑛) 𝕋 𝑏 𝑏
System A: 𝑦 𝐴 𝑛 =𝕋 𝑎 𝑥 1 𝑛 +𝑏 𝑥 2 (𝑛)
System B: 𝑦 𝐵 𝑛 =𝑎𝕋 𝑥 1 𝑛 +𝑏𝕋 𝑥 2 𝑛
Only if system 𝕋 is linear, 𝑦 𝐴 𝑛 = 𝑦 𝐵 (𝑛)

14. Digital Signal Processing Systems: Some Key Properties
Is the system shift variant or not:
If input 𝑥(𝑛) is delayed by 𝑘 samples, what happens to the output?
If system 𝕋 is shift-invariant, output remains the same, except that it also gets delayed by the same number of samples; i.e., 𝕋 𝑥(𝑛−𝑘) =𝑦 𝑛−𝑘
However, if it is not, the output no longer remains the same; i.e., 𝕋 𝑥(𝑛−𝑘) = 𝑦 𝑘 (𝑛)
System behavior for different shifts 𝑘 is different
Question: are the following two systems linear? Are they shift invariant?
System P , given as p 𝑛 =𝑥 𝑛 +𝑥(𝑛−2)
System Q, given as 𝑞 𝑛 =𝑛 𝑥 2 𝑛 + 𝑛−2 𝑥 𝑛

15. Digital Signal Processing Systems: Some Key Properties
Is the system causal?
For causal systems, the output 𝑦( 𝑛 0 ) depends only on 𝑥 −∞ ,…,𝑥 𝑛 0 −1 ,𝑥( 𝑛 0 ). Future inputs, i.e., 𝑥 𝑛 0 +1 ,…,𝑥(∞) do not have any influence on the current output 𝑦( 𝑛 0 )
Is the system stable (more specifically, does it have Bounded Input Bounded Output (BIBO) property )?
For BIBO systems, if 𝑥(𝑛) remains finite, i.e., 𝑥(𝑛) ≤ 𝐵 𝑥 <∞, the output also remains finite, i.e., 𝑦(𝑛) ≤ 𝐵 𝑦 <∞
For unstable systems, the output blows up
A system of practical use is typically both causal and stable

16. Impulse Response of Linear Shift Invariant Systems
Recall definition of unit impulse: 𝛿 𝑛 = , 𝑛= , 𝑛≠0
Let’s denote the output 𝑦 𝑛 =𝕋 𝛿(𝑛) as the impulse response of the system ℎ(𝑛)
Example: for a certain system 𝕋, the output 𝑦(𝑛) when a unit impulse is applied at 𝑛=0 is 𝑟 6 (𝑛−6)
For this system, the impulse response ℎ(𝑛) is 𝑟 6 (𝑛−6)
Why did we introduce a delay of 6 samples in the rectangular window function 𝑟 6 (𝑛)?
𝑟 6 (𝑛−6)
𝛿(𝑛)
𝑥(𝑛)
y(𝑛) 𝕋
𝑛=0
𝑛=0 13

17. Output of Linear Shift Invariant Systems as a Function of Impulse Response and the Input
It turns out that the impulse response ℎ(𝑛) provides all the information needed to compute the output of the system for any arbitrary input 𝑥(𝑛) − provided the system is linear and shift invariant (LSI)
To see this, express any input 𝑥(𝑛), which is a D-T series specified as a function of sample index 𝑛, as a function of two variables; sample index 𝑛 and delay 𝑘
𝑥 𝑛 = 𝑘=−∞ ∞ 𝑥(𝑘)𝛿(𝑛−𝑘)
Note that this is a trick – 𝛿(𝑛−𝑘) remains zero for all the infinite values of 𝑘; except it becomes unity valued when 𝑘=𝑛, at which point 𝑥 𝑘 =𝑥(𝑛).
Why to introduce the complication?  by writing any arbitrary input 𝑥(𝑛) as a weighted sum of delayed impulses, one can write the output (due to the properties of linearity and shift invariance) as weighted sum of delayed impulse responses
System output can be written as follows: 𝑦 𝑛 =𝕋 𝑥(𝑛) =𝕋 𝑘=−∞ ∞ 𝑥(𝑘)𝛿(𝑛−𝑘)
Since 𝕋 is a linear system, by the principles of superposition and homogeneity, the output can be written as: 𝑦 𝑛 = 𝑘=−∞ ∞ 𝑥 𝑘 𝕋[𝛿 𝑛−𝑘 ]
Furthermore, since 𝕋 is a shift invariant system, 𝕋[𝛿 𝑛−𝑘 ] is just the impulse response delayed by 𝑘 samples, i.e., ℎ(𝑛−𝑘)

18. Output of Linear Shift Invariant Systems as a Function of Impulse Response and the Input
Therefore, the output can be written as follows:
𝑦 𝑛 = 𝑘=−∞ ∞ 𝑥 𝑘 ℎ 𝑛−𝑘
This formula represents the convolution operation between two D-T sequences 𝑥(𝑛) and ℎ(𝑛)
There are several ways to understand the convolution operation.
Use the principles superposition and homogeneity
Flipping the impulse response and slide it over the input 𝑥(𝑛): demo on next several slides
From:
Analogy with multiplication of two polynomials
Linear algebra and vector dot product