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Transmit pulse shaping filter – rectangular and sinc (Nyquist)

In the previous post on I-Q modulator and de-modulator, we had briefly mentioned that the a baseband PAM transmission can be modelled as

$s(t) = \sum_{m=-\infty}^{\infty}a_mg(t-mT)$ , where

$T$ is the symbol period,

$a_m$ is the symbol to transmit,

$g(t)$ is the transmit filter,

$m$ is the symbol index and

$s(t)$ is the output waveform.

In this post, the objective is to understand the properties of the transmit filter $g(t)$ i.e. to find out a filter which occupies the minimum required bandwidth while ensuring inter-symbol-interference (ISI) free transmission of the information symbol $a_m$ .

Continue reading “Transmit pulse shaping filter – rectangular and sinc (Nyquist)”

IQ modulation and demodulation

Post describes about the need for I-Q modulation by comparing the spectral efficiency of passband PAM and passband QAM.

PAM (Pulse Amplitude Modulation) is one of the simplest ways to send digital data. In this post, we start with baseband PAM, look at its spectral efficiency, and then move on to passband PAM — which turns out to be less efficient. To fix that, we explore how QAM (Quadrature Amplitude Modulation) improves efficiency by using both sine and cosine carriers. This leads to I/Q modulation, and the corresponding IQ demodulation with Python code.

The content is based onSec 5.2.1 of [DIG-COMM-BARRY-LEE-MESSERSCHMITT].

Table of Contents

Baseband PAM transmission

Consider a simple baseband transmission where the information is sent by modulating a pulse. This can be represented as

$s(t) = \sum_{m=-\infty}^{\infty}a_mg(t-mT)$ , where

$T$ is the symbol period,
$a_m$ is the symbol to transmit,
$g(t)$ is the transmit pulse shaping filter,
$m$ is the symbol index and
$s(t)$ is the output waveform.

Pictorially the same can be represented as,

Figure: Baseband PAM transmission

Minimum bandwidth for ISI free transmission

According to the Nyquist zero ISI criterion, to transmit data without intersymbol interference (ISI), the pulse shaping filter $g(t)$ must satisfy,

$g(nT) = \begin{cases} 1, & n = 0 \\ 0, & \text{otherwise} \end{cases}$

This ensures that each symbol appears cleanly at its own sampling instant without interference from neighboring symbols. For this condition to hold and still remain bandlimited to $\frac{1}{2T}$ , the ideal pulse shape is the sinc function:

$\begin{array}{lll} g(t) &=& \dfrac{\sin\left(\pi t / T\right)}{\pi t / T}, \quad t \in (-\infty, +\infty) \\ &=& \mathrm{sinc}\left(\dfrac{t}{T}\right) \end{array}$

This result comes from the fact that, with ideal sinc pulse shaping, the entire symbol stream can be perfectly reconstructed if the channel passes all frequencies up to $\frac{1}{2T}$ . This is also consistent with the Nyquist–Shannon sampling theorem, which states that to fully reconstruct a signal sampled at $f_s$ , the signal must be bandlimited to $\frac{f_s}{2}$ .

Thus, if we use ideal (bandlimited) pulses like sinc functions, the entire baseband PAM signal can be confined within:

$\text{Bandwidth} = \frac{1}{2T} \text{Hz}$

Figure: Spectrum of baseband PAM (with symbol rate $\frac{1}{T}$ )

Spectral Efficiency

Spectral efficiency is a measure of how efficiently a communication system transmits data over a given bandwidth. It is defined as the number of bits transmitted per second per unit bandwidth, typically expressed in bits/second/Hertz.

$\text{Spectral Efficiency} = \frac{\text{Bit Rate (bits/sec)}}{\text{Bandwidth (Hz)}}$

Assuming that PAM signal $a_m$ uses $N$ distinct amplitude levels, each symbol carries $\log_2(N)$ bits. When each symbol takes $T$ seconds to transmit, the bit rate of the system is:

$\text{Bit Rate} = \frac{\log_2 N}{T} \quad \text{bits/second}$

Substituting the bit rate and bandwidth into the formula for spectral efficiency, we get the spectral efficiency of baseband PAM as.

$\Gamma_{\text{BB-PAM}} = \frac{\log_2 N}{T} \cdot \frac{1}{$\frac{1}{2T}$} = 2 \log_2 N \quad \text{bits/second/Hertz}$

Code

Python code for passband PAM transmission with pulse shaping by rectangular filter and sinc filter.

Passband PAM transmission

Now, consider that the baseband PAM signal is upconverted to a carrier frequency $f_c$ by multiplication by a cosine carrier. The transmitted passband signal is:

$\hat{s}(t)=\sqrt{2}\cos$2\pi f_c t$\sum_{m=-\infty}^{\infty}a_m g\(t-mT)$

The spectrum of the passband PAM is as shown below,

Figure: Spectrum of passband PAM

When the baseband signal is multiplied by $f_c$ , this modulation shifts the entire spectrum up to be centered around the carrier frequency $f_c$ .

The frequency range of the signal becomes:

$f_c - \frac{1}{2T} \) \text{ to } \( f_c + \frac{1}{2T}$

Thus, the total bandwidth needed by passband PAM is,

$\text{Passband Bandwidth} = \left(f_c + \frac{1}{2T}\right) - \left(f_c - \frac{1}{2T}\right) = \frac{1}{T}$

Spectral Efficiency of passband PAM

As earlier, assuming that PAM signal $a_m$ uses $N$ distinct amplitude levels, each symbol carries $\log_2(N)$ bits. When each symbol takes $T$ seconds to transmit, the bit rate of the system is:

$\text{Bit Rate} = \frac{\log_2 N}{T} \quad \text{bits/second}$

Substituting the bit rate and bandwidth into the formula for spectral efficiency, we get the spectral efficiency of passband PAM as,

$\Gamma_{\text{PB-PAM}} = \frac{\log_2 N}{T} \cdot \frac{1}{$\frac{1}{T}$} = \log_2 N \quad \text{bits/second/Hertz}$

Can see that spectral efficiency of passband PAM is half of baseband PAM.

IQ Modulation (QAM for improving spectral efficiency)

As seen earlier, passband PAM suffers from reduced spectral efficiency. To improve spectral efficiency, we exploit the orthogonality of sine and cosine carriers. This leads to Quadrature Amplitude Modulation (QAM), where two independent data streams are modulated:

to a cosine wave (in-phase component, I) $a^I_m$
to a sine wave (quadrature component, Q) $a^Q_m$

Because sine and cosine are orthogonal over a symbol period $\int_T\sin$\omega t$\cdot \cos$\omega t$= 0$ , these two streams do not interfere and can be perfectly separated at the receiver. As a result, QAM doubles the data rate over the same bandwidth compared to passband PAM.

The transmit signal with passband QAM is represented as,

$\bar{s}(t) = \sqrt{2} \cos(2\pi f_c t) \sum_{m=-\infty}^{\infty} a^I_m g(t - mT) - \sqrt{2} \sin(2\pi f_c t) \sum_{m=-\infty}^{\infty} a^Q_m g(t - mT)$

where,

$a^I_m$ and $a^Q_m$ are the in-phase and quadrature data symbols, respectively
$g(t)$ is the transmit pulse shaping filter, and
$f_c$ is the carrier frequency

This signal requires the same bandwidth as passband PAM, however carries twice the information.

Note :

The factor of $\sqrt2$ is included to normalize the total power of the passband QAM signal to unity. As cosine and sine carriers each contribute only half the power $\frac{1}{2}$ when modulating a baseband signal, multiplying each arm by $\sqrt2$ ensures that the modulated in-phase and quadrature components retain their original baseband power.

For notational convenience, we express the passband signal using its complex baseband equivalent,

$\bar{s}(t)=\sqrt{2}\Re \left[ \tilde{s}(t)e^{j2\pi f_c t} \right]$ ,

where

$\tilde{s}(t)=\sum_m \tilde{a}_m g(t - mT)$
$\tilde{a}_m=a^I_m+j\cdot a^Q_m$

This forms the I-Q modulator circuit.

Figure: IQ modulator

IQ demodulation

The received passband signal is split into two branches and multiplied with cosine (in-phase, I) and sine (quadrature, Q) carriers. Each branch is then passed through a matched low-pass filter, typically matched to the transmit pulse shape, to extract the baseband I and Q components. The resulting streams are then independently sampled and demodulated to recover the transmitted symbols.

Downconversion

Let the received passband signal

$\begin{array}{lll}r(t) & = & \bar{s}(t)\\ &=&\sqrt{2} \cos(2\pi f_c t) \sum_{m=-\infty}^{\infty} a^I_m g(t - mT) - \sqrt{2} \sin(2\pi f_c t) \sum_{m=-\infty}^{\infty} a^Q_m g(t - mT)\\ &=&\sqrt{2} \cos(2\pi f_c t) I(t) - \sqrt{2} \sin(2\pi f_c t) Q(t) \end{array}$

where,

$I(t) = \sum_{m=-\infty}^{\infty} a^I_m g(t - mT)$

$Q(t) = \sum_{m=-\infty}^{\infty} a^Q_m g(t - mT)$

To extract the I and Q signals, multiply $r(t)$ with local carrier signals :

In-phase branch:

$\begin{array}{lll} I_{\text{branch}}(t) & = & r(t) \cdot \sqrt{2} \cos(2\pi f_c t) \\ & = & 2I(t)\cos^2(2\pi f_c t) - 2Q(t)\sin(2\pi f_c t)\cos(2\pi f_c t) \\ & = & 2I(t) \left( \frac{1}{2} + \frac{1}{2} \cos(4\pi f_c t) \right) - 2Q(t) \cdot \frac{1}{2} \sin(4\pi f_c t) \\ & = & I(t) + I(t)\cos(4\pi f_c t) - Q(t)\sin(4\pi f_c t) \end{array}$

Similarly, for the quadrature branch:

$\begin{array}{lll} Q_{\text{branch}}(t) & = & r(t) \cdot \sqrt{2} \sin(2\pi f_c t) \\ & = & 2I(t)\cos(2\pi f_c t)\sin(2\pi f_c t) - 2Q(t)\sin^2(2\pi f_c t) \\ & = & 2I(t) \cdot \frac{1}{2} \sin(4\pi f_c t) - 2Q(t) \left( \frac{1}{2} - \frac{1}{2} \cos(4\pi f_c t) \right) \\ & = & I(t)\sin(4\pi f_c t) - Q(t) + Q(t)\cos(4\pi f_c t) \end{array}$

After downconversion, each branch (I and Q) contains the desired baseband signal components, along with unwanted high-frequency terms centered around $2f_c$ .

Low-pass Filtering

To isolate the desired baseband signals, we apply a low-pass filter (LPF) to both the I and Q branches. This filter passes frequencies within the baseband bandwidth (usually $[{-1}/{(2T)}\text{ to } {+1}/{(2T)}]$ where is the symbol period .

Matched Filtering

After low-pass filtering, we apply a matched filter to each of the I and Q branches. The matched filter is designed to maximize the signal-to-noise ratio (SNR) at the symbol sampling instants. It is typically a time-reversed and conjugated version of the transmit pulse shape $g(t)$ .

If the transmit pulse was a rectangular or truncated sinc or raised-cosine shape, the matched filter is also a rectangular or truncated sinc or raised-cosine, respectively. This ensures:

Maximum energy concentration at symbol instants
Minimization or elimination of inter-symbol interference (ISI)

The output of the matched filter is then sampled every seconds (the symbol period) to extract the transmitted symbol values $a^I_m$ and $a^Q_m$ .

This sequence completes the demodulation chain for QAM and ensures that both spectral shaping and ISI suppression are achieved in the receiver.

The IQ demodulator can be visualized as shown in the figure below

Figure: IQ demodulator

Code

Python code for passband qam with rectangular and sinc pulse shaping filter, followed by low pass filtering and matched filtering.

When compared to the transmit symbols, can see that there is a spread in the received symbols. Using raised cosine or root raised cosine filters is a potential approache to mitigate this.

Summary

The post covers the following key aspects

Spectrum of Baseband PAM with a code example
Reduced Spectral Efficiency of Passband PAM
Sending information on orthogonal sine and cosine carriers to improve efficiency (passband QAM)
IQ modulator and corresponding demodulator for Passband QAM with code

Another approach for improving the spectral efficiency of passband PAM is to filter away half the bandwidth (which is not carrying any ‘extra’ information). This is called Single Sideband Modulation (SSB). Anyhow, as I-Q modulation is simpler to implement than circuit for filtering away half the spectrum, I-Q modulation stays. 🙂

Have any questions or feedback? Feel free to drop your feedback in the comments section. 🙂

Reference

[DIG-COMM-BARRY-LEE-MESSERSCHMITT] Digital Communication: Third Edition, by John R. Barry, Edward A. Lee, David G. Messerschmitt

Comparing 16PSK vs 16QAM for symbol error rate

In two previous posts, we have derived theoretical symbol error rate for 16-QAM and 16-PSK modulation schemes. The links are:

(a) Symbol error rate for 16-PSK

(b) Symbol error rate for 16-QAM

Given that we are transmitting the same number of constellation points in both 16-PSK and 16-QAM, let us try to understand the better modulation scheme among the two, i.e. to answer the following question:

For the same signal to noise ratio $\frac{E_s}{N_0}$ , will 16-PSK or 16-QAM give a lower symbol error rate?

Continue reading “Comparing 16PSK vs 16QAM for symbol error rate”

Blog on DSP

I happened to visit ‘The Digital Signal Processing Blog’, maintained by Mr. Andres Kwasinski, Ph. D.

In the blog one can find details about the upcoming IEEE conferences pertaining to communication and multimedia processing.
Further, in some of the posts, author shares his thoughts on topics like fixed point arithmetic (here) and wavelets (here) etc.

Thanks,
Krishna

Symbol Error Rate for 16PSK

In this post, let us try to derive the symbol error rate for 16-PSK (16-Phase Shift Keying) modulation.

Consider a general M-PSK modulation, where the alphabets,

$alpha_{16PSK} =\sqrt{E_s}\left\{1,\ e^{\frac{j2\pi}{M}},\ e^{\frac{j4\pi}{M}},\ \ldots,\ e^{\frac{j2\pi(M-1)}{M}} \right}$ are used.

(Refer example 5-38 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT])

Figure: 16-PSK constellation plot

Continue reading “Symbol Error Rate for 16PSK”

Frequency offset estimation using 802.11a short preamble

From the previous post on OFDM (here), we have understood that an OFDM waveform is made of sum of multiple sinusoidals (also called subcarriers) each modulated independently. In this post, let us try to understand the estimation of frequency offset in a typical OFDM receiver (using the short preamble specified per IEEE 802.11a specification as a reference).

Understanding frequency offset

In a typical wireless communication system, the signal to be transmitted is upconverted to a carrier frequency prior to transmission. The receiver is expected to tune to the same carrier frequency for downconverting the signal to baseband, prior to demodulation.

Figure: Up/down conversion

Continue reading “Frequency offset estimation using 802.11a short preamble”

Peak to Average Power Ratio for OFDM

Let us try to understand peak to average power ratio (PAPR) and its typical value in an OFDM system specified per IEEE 802.11a specifications.

What is PAPR?

The peak to average power ratio for a signal $x(t)$ is defined as
$papr = \frac{\max\left[x(t)x^*(t)\right]}{E\left[x(t)x^*(t)\right]}$ , where
$()^*$ corresponds to the conjugate operator.

Expressing in deciBels,
$papr_{dB} = 10\log_{10}(papr)$ .

Continue reading “Peak to Average Power Ratio for OFDM”

Cylcic prefix in Orthogonal Frequency Division Multiplexing

In a previous post (here), we discussed in brief, Orthogonal Frequency Division Multiplexing (OFDM) transmission. Let us know probe bit more into the motivation of cyclic prefix (aka guard interval) associated with each OFDM symbol.

What is cyclic prefix?
Let us consider one subcarrier (subcarrier +1 specified in IEEE 802.11a specification) alone. In the figure shown below, the blue line corresponds to the original sinusoidal where one cycle of the sinusoidal is of duration 64 samples ( $3.2\mu s$ with 20MHz sampling), corresponding to subcarrier of frequency 312.5kHz.

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Understanding an OFDM transmission

Let us try to understand simulation of a typical Orthogonal Frequency Division Multiplexing (OFDM) transmission defined per IEEE 802.11a specification.

Orthogonal pulses
In a previous post (here ), we have understood that the minimum frequency separation for two sinusoidals with arbitrary phases to be orthogonal is $\frac{1}{T}$ , where $T$ is the symbol period.

In Orthogonal Frequency Division Multiplexing, multiple sinusoidals with frequency separation $\frac{1}{T}$ is used. The sinusoidals used in OFDM can be defined as (Refer Sec6.4.2 in [DIG-COMM-BARRY-LEE-MESSERSCHMITT]:

$g_k(t) = \frac{1}{\sqrt{T}}e^{\frac{j2\pi kt}{T}}w(t)$ , where

$k=0,1,\ldots,K-1$ correspond to the frequency of the sinusoidal and

$w(t)=u(t)-u(t-T)$ is a rectangular window over $[0\ T)$ .

Continue reading “Understanding an OFDM transmission”

Simulating Minimum Shift Keying Transmitter

Minimum shift keying (MSK) is an important concept to learn in digital communications. It is a form of continuous phase frequency shift keying . In minimum phase shift keying, two key concepts are used.

(a) The frequency separation of the sinusoidals used for representing bits 1’s and 0’s are $\frac{1}{2T}$ , where $T$ is the symbol period.

(b) It is ensured that the resulting waveform is phase continuous.

Motivation of continuous phase

In a previous post (here), we have understood that the minimum frequency separation for two sinusoidals having zero phase difference to be orthogonal is $\frac{1}{2T}$ , where $T$ is the symbol period. However, it can be observed that at each symbol boundary, there is a phase discontinuity. The presence of phase discontinuities can result in large spectral side lobes outside the desired bandwidth. Hence the need for having a frequency modulated signal which is phase continuous.
Continue reading “Simulating Minimum Shift Keying Transmitter”